Directors of Amtrak, braced with new statistics showing the impact of inflation on the rail corporation's deteriorating financial status, voted yesterday to seek $56.5 million of additional money from Congress for the next fiscal year.

Amtrak officers said that without the injection of new money, additional cutbacks in routes and service frequency would have to made starting Jan. 1. Such reductions would come on the heels of cutbacks already announced for the Washington-Boston corridor and elsewhere in Amtrak's national system.

The Carter administration's representative on the Amtrak board. Deputy Federal Railroad Administrator Robert E. Gallamore, cast the sole vote yesterday, against seeking a supplemental appropriation.

Gallamore also cast the single vote against adoption by the board of a resolution on long-term policy for the government-subsidized rail passenger operaion. The majority supported what was described as an "expansionist approach" for the years 1979-1982, under which federal subsidies would be maintained at current levels with excess revenues being applied to improved service.

A spokesman for Secretary of Transportation Brock Adams, long a supporter of Amtrak, reaffirmed the secretary's enthusiasm for "intercity passenger service" but said Adams thought "the amount requested was unreasonably high."

Adams is scheduled to meet on Wednesday with representatives of the National Association of Railroad Passengers, which has denounced recent service cutbacks by Amtrak and declared that rail passenger service "is in greater danger now than ever before."

Earlier this year, Carter administration officials endorsed a policy under which service would be reduced if increases in Amtrak revenue fail to offset inflation.

Congress has authorized $545 million in taxpayers' subsidies to cover Amtrak losses for the fiscal year starting Oct. 1 but seldom matches authorizations with actural appropriations for any government program.

In the case of Amtrak, Congress appropriated $488.5 million for operating subsidies - $56.5 million below its Amtrak's original request for $534.1 million and $11.5 million below the $500 million level supported by President Carter.

Because of the losses projected under a $488.5 million ceiling on its deficit, Amtrak has announced a series of unprecendented service cutbacks to start Oct. 30, including elimination of 22 trains a day (out of 120) on the heavily patronized Boston-Washington corridor.

Coupled with several earlier reductions of service, the recent decision by Amtrack's officials would reduce the deficit by some $28 million. However, since the overall projected shortfall in the new fiscal year is $56 million, officials said yesterday that our train reductions must take place starting in a few months.

Yesterday, Amtrak officers revealed to the directors that a hiring freeze was instituted Sept. 1, accompanied by a goal of reducing total employment by 6 per cent in the coming year.

In addition, several members of the board expressed surprise when told that new financial projections for the coming fiscal year show $11 million increases in labor and materials costs that were not anticipated in a forecast made last December.

Specifically, Amtrak labor officials now expect a new settlement with rail labor unions to add nearly 11 per cent in costs over the current fiscal year compared with a rise of 9.64 per cent expected last December.

Thus, even if Congress approaves some additional funding for Amtrak, it is uncertain now much money would be available to restore services already earmarked for elimination. Amtrak president Paul Reistrup told the board yesterday that if the full $56.5 million of additional money is appropriated, most of the cutbacks now planned could be eliminated.

If Congress agrees to appropriate the additional $11.5 million needed to reach the level of spending previously endorsed by the administration, that would mean no new cutback. But reductions in service already announced would continue because of the new inflation forecast.

A check with aides to Democratic and Republican members of Congress yesterday revealed no enthusiasm for additional Amtrak appropriations.

Amtrak officials will meet Wednesday with staff members of the House Appropriations Subcommittee on Transportations, headed by Rep. John J. McFall (D-Calif.) to review recent dicisions on service cutbacks. Even if the McFall subcommittee endorses additional funds, sources said it is unlikely that the full committee, headed by Rep. George H. Mahon (D-Tex.), would agree to additional spending without pressure from the administration.

In other action yesterday, meanwhile, Amtrak's board defered until Sept. 28 a decision on whether to ask Congress also for additional capital improvement spending. Congress has authorized $130 million for the coming year but has appropriated $108 million.