John Coltrane and Miles Davis perform during Coltrane’s final tour with the Miles Davis Quintet. ( Courtesy of Trapeze Music & Entertainment Limited)

When the Miles Davis Quintet headed to Europe in spring 1960, John Coltrane had already submitted his resignation. If he felt the common dread of leaving something big behind, it didn’t show: Throughout this tour, Coltrane hit on a kind of unvarnished intensity and seemed to be playing almost in spite of the rhythm section. His career had reached full adulthood, and he was adjusting to a new set of powers.

Recordings from those 1960 concerts have trickled out over the years via small European labels, culled from radio broadcasts or private bootlegs. “All of You: The Last Tour 1960,” a new four-CD box of remasters released on Britain’s Acrobat imprint, is the mother lode, the fullest composite available. The eight performances included here show an artist apart, drifting into himself.

“He was in stark transition, and he was under-recorded during that time,” said Andrew White, a Washington saxophonist and scholar who has transcribed more than 800 Coltrane solos. “He was especially fertile on that tour with Miles. There are strong shades of it on the ‘Kind of Blue’ record in ’59, but it was on that tour that he shocked everybody.”

The band moves snugly, speaking in its familiar, crowd-pleasing musical jargon. Coltrane, meanwhile, spends long passages squeezing brittle split tones from the saxophone, or spinning circular, warbling phrases beyond exhaustion.

“I think he plays on that tour differently from how he plays on any other recording,” said the saxophonist Simon Spillett, who assembled the collection. “You can hear the seeds of where he’s heading. He’s pushing the saxophone and its context to its limits. Already in his head he was hearing a very different accompaniment from what [pianist] Wynton Kelly and [drummer] Jimmy Cobb were giving him.”

In the late 1950s, the iconic version of Davis’s quintet turned the trumpeter into jazz’s leading man and crowned Coltrane as the tenor saxophonist to watch. This worked partly by way of contrast: Davis was the slick, pretty-faced enigma while Trane the ungainly, sweater-wearing intellect.

But after five years, Coltrane’s concept was too big and implacable to play the foil’s role. In January 1960 he’d released “Giant Steps,” a game-changing album of swiftly restructuring harmonies and fast, staggering solos. The next year, he ran just as hard in the opposite direction, releasing “My Favorite Things” and “Africa/Brass,” stern and formidable records where the chords hardly moved at all. By the time he established his so-called classic quartet in 1962, he was focused on building resolute foundations. The band drew on free jazz pioneer Ornette Coleman’s influence, plus Coltrane’s own studies of Indian and West African classical musics, and helped redefine the jazz combo as a breathing, unified body.

But in March 1960, Davis could find no suitable replacement for him, so Coltrane agreed to make one final trip. On the first night of the tour, at the Paris Olympia, startled boos mingled with applause during Coltrane’s solos. That show was left off the “All of You” box because of rights claims, but even in front of northern Europe’s more cordial audiences, a sense of restlessness is easy to detect. As his solo on “So What” peaks into ungraspable flurries, or his off-scale playing on “All of You” pulls Cobb off his accents, listeners must be wondering — is this the guy we’ve been listening to on “Kind of Blue”?

“All of You” catches Coltrane at the beginning of what can be called his eruptive period, full of rapid-fire innovations. The music went inward, but distilled much of what was going on around him: nuclear testing; the fiery launch of the first American space probe; grenades tossed in the marshes of Vietnam; conflagrations of entreaty, ignited in American slums.

This era began with Coltrane’s decision to reorient Davis’s conception of “modal jazz” — that is, pieces that stay on a single chord for an entire section — to his own ends. On Davis’s “Kind of Blue,” jazz’s most famous modal album, Coltrane and the other soloists glide above the rhythm section, chasing melodies into air. On “All of You,” Coltrane’s improvised patterns are rough — units of rhythm more than anything else, that slice the band’s sense of continuity.

He’d always built his solos around small motifs, but the wide-open harmonic system of modal music was new: It seemed to give him full control over how and when he made the motifs, while raising the improviser’s stakes. Throughout “All of You” he centers on one melodic idea, approaches it from various angles, exhausts it, then moves on to another.

“I start from one point and go as far as possible. But, unfortunately, I never lose my way,” Coltrane told an interviewer in January 1962. “I ‘localize,’ which is to say that I think always in a given space. I rarely think of the whole of a solo, and only very briefly. I always return to the small part of the solo that I was in the process of playing. The harmonies have become for me a kind of obsession.”

On albums such as “Blue Train” and “Giant Steps,” Coltrane had gone to an extreme, making tunes that changed not only their chords but their entire keys every few beats. When he embraced modal composition starting in 1960, the common wisdom was that he had exhausted the potential of chord movement — there was just nothing else left to do.

But there are always new means of assembly, especially for a scientifically minded creative inventor like Coltrane. In questioning the need for complex structures, he was working toward a kind of interrogative beauty that engulfed as well as challenged. On “All of You,” his repeated motifs and incantatory helixes of melody are exercises, aimed at perfecting a method of inclusive prayer. By 1963, when he made “Live at Birdland” — including a burning, 13-minute rendition of “Afro Blue” and a tears-and-sweat take on the ballad “I Want to Talk About You,” ending in three minutes of unaccompanied saxophone — he had condensed his powers: He could evoke that feeling of communion in just a single swipe of notes.

The next year, when he recorded the modal and driving “A Love Supreme,” he could bottle the effect into concise, three- or four-note “cells.” At a November 1966 performance in Philadelphia, dug up and released this year as “Offering: Live at Temple University” (Impulse!/Resonance), he occasionally sneaks these cells into resolutely freeform playing, hitting one of his career’s stunning highs. (Less than a year after the Temple concert, Coltrane died at age 40.)

“When you remove the template, you’re now the one creating the structure. In his playing, that’s completely deep,” said the saxophonist Logan Richardson, 34. “We could talk about the harmonic structures and the chords and the architecture of how he was able to put everything together while also being the heavyweight champion, able to peak and peak and peak. But I think just the constant search and the hunt — that in itself is the focus with him.”

It’s a concept of critical importance to Richardson’s generation, which is adjusting to a new kind of modal playing now that funk and hip-hop sampling have redefined jazz. Today, jazz drumming centers on the bass drum and snare, not the cymbals; harmonic motion is slower, more viscous. The jazz solo never fully moved on from Coltrane’s influence, but with a new movement afoot the question becomes more clear: Will anyone achieve the transcendental jazz solo in the post-hip-hop age, as Coltrane did when global and avant-garde influences infiltrated the music in the early 1960s?

The drummer Chris Dave seemed to be hinting at the idea on his 2013 mixtape, Chris Dave and the Drumhedz. On the opening track he included a reworked, sample-like version of Coltrane’s famous incantation from “A Love Supreme,” intoned over a loose hip-hop beat. But the problem of Coltrane’s legacy, which becomes perhaps the main problem of jazz in the past 50 years, is that he was building a personal (though openhanded) system of divination, not an easily-replicated strategy. Those looking to take it forward will have to inquire within.

Russonello is a freelance writer.