Id think the .223 would be the more obvious choice, can still find one in a heavy barrel if you like, for now anyway
but, the obvious choice is seldom the most fun so, heck, .357 is good too, although you may hate it once that sucker hangs up on you past your comfort zone.
you may want to go with the .30-30 very versitile, and cheap if you dont reload, plus in your sig you have one already so that is a plus, can reload for both.
Now, this is what I have done. I got a .223 for the long shots, and a 12ga pard for the close ups, I just got the 10ga turkey as well if Im feeling strong that day. Im now looking to get the TDC just so I have one, just in case you know something happens, like looking in the locker and realizing I dont have one.
good luck to ya
Dave