The office of Kentucky Governor Steve Beshear just announced that more than 370,000 people have now signed up for Obamacare on KyNect, the state exchange. More than one out of every dozen Kentuckians — 8.6 percent of the state population — now has obtained coverage through the exchange, Beshear’s office said, claiming that a preliminary analysis has established that three out of four enrollees has reported that they were uninsured before signing up.

Beshear’s office adds that more than 21,000 signed up in the last three days alone.

But as recently as three days ago, Senator Mitch McConnell of Kentucky was denouncing Obamacare as “disastrous,” lamenting the “catastrophic effects” the law has had on Kentucky families, and insisting that “the pain caused by this terrible law is easy to see.” He claims the law must be uprooted “root and branch.”

So the question is, Does there come a point at which Alison Lundergan Grimes can more directly target McConnell for wanting to take health coverage and security away from hundreds of thousands of people?

Grimes has generally stood up for the idea of expanding health coverage to Kentuckians and has said we should not repeal the law. But in a recent interview with Philip Rucker, she quickly moved off the topic of Obamacare, as is her wont.  And some Kentucky Democrats, such as Rep. John Yarmuth, have suggested that there are ways Grimes might be able to take on McConnell over repeal a bit more aggressively, particularly given that enrollment numbers continue to mount in Kentucky, expanding coverage to untold numbers of poor, rural folks who desperately need it.

Grimes appears to be adopting a strategy of deliberately avoiding engagement on Obamacare. There’s a plausible rationale for this. The campaign seems to have calculated that above all, she should avoid getting drawn into Washington arguments (such as the one over the health law). That’s because one of her chief strengths is that she never served in Washington (and didn’t vote for the law). If Grimes’ main assets are that she’s very much of the state and represents something new — making it easier to paint McConnell, with his low approval ratings, as part of the Washington problem — then avoiding association with Washington Democrats and controversial national legislation perhaps makes sense.

What’s more, Dem Governor Steve Beshear — perhaps the most aggressive defender of Obamacare in the south — can do the heavy lifting in terms of attacking McConnell for wanting to take coverage away from hundreds of thousands. He tends to speak of the law’s achievement in Kentucky in strong moral terms, as a great policy achievement that has helped people’s friends, neighbors, co-workers, and fellow church-goers. While he does this, one line of thought has it, Grimes is free to focus on the minimum wage, equal pay, and her jobs plan for Kentucky.

Here’s something to watch for along these lines. While “Obamacare” and “Obama” are still toxic in Kentucky, the exchange known as KyNect appears to be enjoying a far better reception in the state. As Perry Bacon has pointed out, “Kentucky’s implementation of the Affordable Care Act has been wildy successful,” so much so that even Republicans in the state speak highly of the exchange:

Even Republicans here say that some Kentuckians will criticize Obamacare but in the next breadth emphasize how well KyNect works, as if they are not part of the same law.

So it wouldn’t be surprising if you saw Grimes start hitting McConnell for wanting to repeal KyNect, especially if enrollment mounts and a critical mass is reached. After all, even Republicans in the state tend to talk about it as a Kentucky creation, not a Washington one.