That would be nice, but...
...probably not. The US hasn't had a President and Vice President from different parties since, I think, John Adams was President. Thomas Jefferson was his VP. It was a four-year political dogfight, and nearly nailed the coffin shut on what had been a warm friendship earlier in their lives.
If Obama asked him and the Democratic Party consented, I doubt Hagel would accept. He has already said he will not run for another term in the Senate and will not be a candidate for any office in 2008.
I do wonder, though, if Hagel would accept a Secretarial appointment to Defense, if Obama were elected and if it were offered. Cabinet appointments across party lines, while rare, are not unheard of.