I had a Billy Beagle (house pet) for years (he died in Nov 2011). Now I have Rescue Millie Beagle, 1.5 years old. She is definitely a hunter. I turn her out in the yard in the morning and she goes to town. I realize my Billy was dumped because he wasn't interested in hunting. He was just a wonderful, sweet boy. I have spent almost a whole week making my chain link fence (about 1/3 acre yard) Millie-proof. She is going to be a character, but I already love her. She sleeps on the bed beside and at night, then goes out to 'hunt' in the morning.
Sorry about your loss of your pal Billie. Millie sounds like she will keep you happy and busy! One of my beags used to be a hunter and it was a challenge at first to get her used to being in a fenced yard and on a leash, but she is super happy now.