This would seem to be a matter of conjecture until such time, if ever, that The Supreme Court ever renders an opinion on this issue. In the meantime, here is my WAG:
There is nothing in the 22nd Amendment which explicitely prohibits a former president from later holding another Federal office, whether by election or appointment. While modern presidents have not sought such positions, from time-to-time prior to the passage of the 22nd Amendment former presidents did hold federal positions: William Howard Taft was Chief Justice of The Supreme Court. Andrew Johnson was a Congressman. John Quincy Adams was not only a Congressman, but served as Speaker of the House. There may be other examples.
I can’t picture that Clinton would be barred, say, from being appointed to the Cabinet (not that this seems likely to ever happen), or from running for Congress, because he would be placed somewhere low on the order of succession, and it would be remotely possible that he would become president again if a slew of people died.
By the same token, I don’t think he would be barred from becoming Vice President either. If he were, and the President died with less than two years left in his term, there would be no bar to Clinton serving as president for the remainder of the term.
And if the President died with more than two years left to his term? My WAG is that he would succeed to the presidency, but he would be considered to have his term of office expire at the end of two years of service. The 22nd Amendment would then be invoked again, this time to appoint his successor for the remainder of the original President’s term.
This means that there would be a 50-50 chance that, if the President died, his term would have to be served out by at least two people. There would, no doubt, be a good deal of discussion and objection by politicians and the electorate alike to the potential of exposing the country to this kind of confusion and discruption, and, for this reason if none other, I expect any presidential candidate would be dissuaded from naming a former two-term president as his or her running mate.