I'm not going to say FDR was the most recent well-liked but he was extremely well liked when he was president. He got voted into the most terms were (4). He also would talk to the American people over the new invention of the radio. The American people would gather to listen to his voice which was considered a great radio voice. Went he died his body was moved by train and as the train passed cities and towns people saluted. Nowadays people are more likely to criticize him for the Japanese internment camps but during his administration he was a loved president. A problem with your overall question is all presidents have legitimate criticisms to them including the most loved George Washington and Abraham Lincoln.

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It's a good question that i'm not going to be able to fully answer (and it seems to me the best answers would have to dive deep into <20 year history) but reducing this question to a more simplistic one can provide some answers. "What presidents were well liked by the opposing party" and gallup has a nice resource for that. This goes from Truman to Obama

George HW bush polled above 50 percent with democrats from june 1989 to september1991 (there is a 100 day period right before desert storm where he falls under before shooting back up to above 70 percent during the war)

Kennedy spent over half his term with his popularity over 50 percent among the GOP but his opinion is noticably down during the end of his turn.

but the real clear cut universally well-liked figure is Eisenhower:

you can also look at this another way: pre Bush-Obama how many presidents have had opposing favorabilities so low and you really only get Harry Truman and the late late Watergate Nixon (and even he resigned with 11% favorability and Carter in the last two quarters of his presidency. Complicating the "well liked" argument above George HW Bush fell to under 20 percent for pretty much the whole of 1992.

hope this helped.