I'd go for (another!) M1 Garand myself.
There seems to be a good supply of them at most good sized gun shows. Though most are over priced,,they don't seem to sell at those prices either.

A decent one can usually be bought yet for betw $500 & $700. Do some homework on what to look for and then do some shopping. Plenty of parts available for maint.

As far as using it for deer hunting, 5 shot enbloc clips are available to meet those regs. MilSurp M1 30-06 ammo can be used if you pull the FMJ bullets and reseat with commercial softpoints. Saves alot of reloading hassel for a few hunting and sighting in rounds.

The older Remington commercial semiauto rifle (M742??) is good suggestion also. They usually go begging for owners on gunracks.
The Browning commercial BAR semiauto would be a nice one too. There's one on one of the auction sites now in 30-06 for under $800.

For the cool factor, you could go with a Remington Model 8 or 81. Either in cal.35 Remington,,or the latter in 300 Savage