The reason I always liked the .35-caliber, is that the .357"-.358" bore seems to be a perfect diameter for a cast-bullet deer cartridge. A 200 to 250 grain gas-checked .358" cast bullet pushed along at a moderately quick speed has plenty of oomph at close range, and retains it better downrange than a larger bore, unless you are willing to push a really heavy .40 or .45 caliber bullet to a similar speed. Then you end up with that old bugaboo, a flinch. At least, I do.
The .35 Remington is a perfect case for this use, and the various .35/.30-30 wildcats meet the criteria as well. The old .35 WCF and the .35 Whelen would be at the top end of this category, and the .357 Maximum is at the bottom. My .357 Max Handi is a real mild rifle to shoot. NEF seems to have only two barrel contours for center-fire rifles (except for bull barrels), and .35 calibers are made in the heavier contour, giving the rifle a somewhat muzzle-heavy balance, and very mild recoil. That much extra steel would be quite welcome, however, in a Handi chambered for .35 Whelen, .35 WCF, or even .356 or .358 Win.
A Handi with a 24" barrel, chambered in .35 WCF, equipped with good-quality open sights, perhaps a quarter-rib rear sight base that also serves at a scope mount, fitted with a classic black walnut stock, simple checkering . . . some talented amateur gunsmith could make a fine deer gun from a plain old Handi, while snowed in over-winter.