For cheap shooting I use a 22K hornet , then its a choice of: 204,223,22-250,22Cheetah,243,6mm Rem, 6mm Ack Imp,250 Savage,257 Ack Imp,25-06, 25-06 Ack Imp,7 TCU,7-08, 270 Win.357 Max
The Rooster is partial to his 357 but I have a 220 Swift in the works for him.
To be honest though not all of my choices are in a Handi, but 8 of them are.