Jon Bernstein had a post recently asking whether President Clinton moved toward the center following the 1994 election. Bernstein largely comes down on the side of no, although he cites a Brendan Nyhan post offering evidence that Clinton did moderate somewhat.
Actual quantitative evidence of presidents' ideological dispositions (no less shifts in them) is difficult to come by. For what it's worth, I was working for Clinton during this election cycle. While my low-level job was far removed from the Oval Office, in terms of both geography and power, there was a sense of moderation felt throughout the office in the wake of the '94 midterms.
One of the major changes at the time was really one of tone. Mack McLarty, the chief of staff who was close with the Clintons and whom all the Arkansans knew and liked, was out, replaced by centrist Leon Panetta from California. And a signal was sent (although I can't recall a specific mechanism) that it was time to get serious. My office in Correspondence had been quite a fun place to work prior to the '94 election. We worked hard, but the environment had been one more like that of a campaign than a government office -- very informal, a bit rowdy. Pretty quickly after that, the office environment changed. It became less fun, more professional. I stopped wearing Star Trek and Elvis ties. (Seriously, I did. Granted, I probably shouldn't have been wearing them in the first place.)
Now, only in the broadest, non-political sense does a making a chaotic office more professional constitute a rightward shift. But I also had a sense of movement toward more conservative public policies.
As a letter writer, one of the issue areas I dealt with was health care. I recall working on a letter in early 1995 related to the treatment of drug abuse as a public health problem. I wish I could remember the specifics of the policy, but there was a notable shift between 1994 and 1995, where the White House was no longer supportive of providing certain funds to help those dealing with drug addiction. This sticks in my memory because I remember double-checking with the Office of Domestic Policy about it, as it seemed like we were adopting the Republican position. I then had to check yet again when my supervisor didn't believe what I'd written in the letter. I believe there were other modest shifts in policy around then, although nothing that would generate major headlines and nothing that sticks out in my head.