If you can find one, a .250 Savage kicks less than a .243, and comes within 50 f/s in velocity with a 100 gr bullet, as the .243 with a 90 gr. bullet. The .243 is a good choice if you can't find the Savage, which could take some time, and is limited in available factory ammo. If you handload, the .250 is a great cartridge.
The .260 is a good round, and is more versatile than the .243, but with heavier bullets does kick somewhat more than the .243. I have a Ruger M77 MK II Compact with a 16.5" barrel, and with the 100 gr nosler Partition I can't tell the difference between it and the .243, but the blast from the short barrel is rather intense. My wife did not like that aspect of the Compact.
The .25-06 is another good cartridge, but it does kick more than the .243, .250 Savage, .257 Roberts or the .260. If you're looking for a low recoiling .25 caliber with readily available ammo and a factory rifle, get the Ruger 77 MkII Ultra Light in .257 Roberts. It is an outstanding cartridge, in a handy rifle. I have a friend who uses one. He loves it, and says it is outstanding on Whitetails. I've shot it, and it's a great little rifle. Subjective recoil is similar to the .260.
You didn't say if you handload or not. If you do, you can control the amount of recoil in any caliber by loading light loads or reduced loads. When I was younger, I started deer hunting with a Remington 700 in .30-06. As a 13 year old weighing about 120 lbs. the full house 150 grain loads kicked my young butt. That Christmas I got an RCBS Jr. single stage reloading press with all the other goodies, and at that point I started experimenting with 125 gr bullets, reduced 150 gr loads, and cast bullet loads. It made a huge difference in comfort, as well as increasing the amount of practice in which I engaged. It also helped keep costs down. I could handload three to five more boxes of ammo for the same price as one box of factory ammo.
I'll probably be flamed intensely for this idea, but here goes. Have you considered the .223? With the Nosler 60 gr Partition, or the Winchester 64 gr Powerpoint, it works well on Whitetails, IF you limit your shots to broadside only within 150 yards. Just an idea.