The tip of the tool is concaved and the shaft is 3 1/2" long and about 3/8" in diameter. 

Probably one of these with a handle-

Perhaps for these-

I think you may be correct with that papaw, if the shaft goes thru the handle

Some of the "snap" rivets don't need too much force, so I think a handled tool "could" be sufficient.

Brian, was it sharp? Like a leather tool? If not, maybe a tool for installing push-on nuts like on Tonka Toy wheels? Kitchen tool for making Red-Hots very slowly? Pharmaceutical tool for making Sudafed-like pills in limited volume? Wood handle says pre 1950. If the shank is through the handle, it might be a rivet setter, but the handle should show some damage from striking. None of the rivet tools I have seen have wood handles.

Good guesses so far... It is a hand tool, no hammers involved...
I will say this it can get messy using this tool... 

HOW ABOUT A PITTER ,CHERRY. PEACH

Well...pitting cherries with it *would* be messy.....

Nope that's not it...

No more guesses???

>I will say this it can get messy using this tool...

Does it involve animal reproduction? If so I'm not guessing - P

Not even close Rusty...

So, we know it's not to be struck, so it's more than likely pushed in use.

-Concave end.

-Can be messy in use.

Could it be for carving something soft, in a push type manner so the removed material rolls up and out of the way? Leather? clay sculpture?

Let's see... Yes it is to be used to push something fairly soft, but not leather or clay... George Washington could have used a tool very similar to this one...

Wood dowel pusher tool?

Upholstery tack pusher?