Since you just got it I would say no. If you had it for a long time and practiced with it lots then the story may be different.
During the great depression my now deceased grand father killed many deer with a 22lr. That said you have better options. Most on my deer have been taken with a 357 magnum but I have owned that gun for about 25 years and shot many 1000s of rounds through it. That gun more than any other I own I feel is a part of me. When I hold it, the gun feels as natural as holding nothing.
If It was me with your guns I would use the 30-30. Id use the 30 carbine for coyotes or other similar sized animals.
Shot placement is everything! I would rather take a round of 50BMG to the foot than a 22short to the base of the skull.
all 26 of my guns are 45/70 govt, 357 mag, 22 or 12 ga... I believe in keeping it simple. Wish my wife did as well...