No, if you serve more than two years as President, you're only eligible to be elected for one more term.
That's why LBJ was eligible for reelection in 1968 (he had served under two years of Kennedy's term), while Truman would not have been in 1952 had the 22nd amendment applied to him. (However, it didn't, so Truman was hypothetically eligible to seek two more terms, since the 22nd hadn't yet gone into effect. Eisenhower, however, was ineligible for a third term.)
Does this also mean Herbert Hoover (who lived to 1964!) would have theoretically been able to serve 2 more terms if he had run in 1952 or later? LOL at him getting elected again in the 50's though.
EDIT: No, it looks like the total exemption was written to cover only Truman, because he was president when Congress proposed the amendment. Had it taken decades to ratify, it looks like it would have also allowed a future president to
finish a 3rd+ term in office if they were already elected to one, but not to
run for a 3rd+ term after it took effect.
