Radar's Hat's resting place

Are you sure it's his hat? In his final episode before leaving Korea, he left his teddy bear behind, but I don't think he did anything special with his hat.

He was without his hat for most of his final episode, which was actor Gary Burghoff's idea. He wanted to show that the character of Radar was no longer a kid and was grown up, so he went without his hat, which showed off his receding hairline.

Gary Burghoff decided to not wear his hat for the final episode:

When it finally came time to film “Goodbye Radar” Gary decided he didn’t want to wear his hat. This became a big issue and remains a sore spot with me to this day. Our contention was that without the hat he no longer looked like a kid, he looked like a balding man rocketing into middle age.

No on-screen explanation has been given that I know of.

Perhaps, as PaulStock has pointed out, you're conflating the hat with Radar's teddy bear. From an article on mash4077tv.com:

“Good-Bye Radar (Part II)”
After Radar leaves the 4077th for good, Hawkeye and others return to the Swamp to discover that Radar has left his teddy bear on Hawkeye’s bunk.

...and, from the show's final season, well after Radar had left:

“As Time Goes By”
Hawkeye offers up Radar’s teddy bear, which was left to him: “Let it stand for all the soldiers who came over here as boys and went home as men.” Note that the teddy bear seen in this episode bears little resemblance to the one-eyed bear shown in every other episode.