I have literally seen the results thousands of Texas deer shot with 22 cal centerfire bullets .These deer were shot by kids from children's home under the supervision of seasoned guides (operationorphans.org).The most common cartridge used is the 222 Remington (camp has 21 of them), but other various cartridges are used. The results are very positive, a dead deer and a very happy youngster and believe it or not a happier guide.
I will not hesitate to use a 222, 223, 22-250, or even a 22 Hornet for a
properly placed shot on a Texas Whitetail deer.No they will not have an exit hole but they are DRT or within a few steps so why do you need a blood trail ? As far as bullet choice goes, use one that shoots well in your rifle.One well placed shot and it is time to get out the knife and go to work!
357 in a Handi is a great choice but it will NOT kill a Texas Whitetail any deader than a properly placed 224 bullet. I own a 357 and a 357 max in Handi's , they are great guns but they are not a better deer killer , after all dead is dead
.
A 308 (or any other 308 based case is a good choice) if you desire a larger caliber.
As Pete says Get a bunch and make it difficult to decide which one to carry