Unless you absolutely need a semi auto 22, I would suggest anything else but. And using the words semi auto and "cheap" in the same sentence when discussing 22s is going to mean trouble more often than not.
I had several over the years and don't think I ever got through a box of ammo without a jam,misfire, etc.
I have an older Marlin .22, can't remember the model. You have to keep it clean, or the action does not through the spent casing out right and it jams. I can shoot the center out of a clay pigeon at 50 yards, so they are accurate enough, just a bit troublesome. Now, i would recommend a bolt action, personally. Unless there is some urgent, unresolved matter between you and a "rascally rabbit", I wouldn't go out and get a cheap firearm of any type. Shop around, network a bit and save. Get a good one that you'll be happy with for a while. There is nothing worse than setling for something, then growing to hate it, and knowing full well that was coming. Then, you'll go out and spend the money on a good one anyway. my Dad has a beautiful Remington .22, i've loved it since i was a young kid. I've had an oppertunity to use it on some rabbits, and it was great.
i have to agree with the others that said you cant go wrong with a 10/22 ive had mine four about 5 years now and it still shoots just as good as it did the day i got it and its very accurate. you can get a couple of the models for under 200 bucks i think i paid around 250 for mine but that is with a wood stock and ss barrel. i used mine this morning to get some black birds out of the field and actually hit 1 at 90 yards which impressed me since ive only got it sited in at 50.
just curious what brand ammo were you shooting when it jamed i found that remingtons shoot best out of mine and without many jams at all if any and the few times that i tried federals i had a couple jams. another thing you might want to look at is getting a couple of the butler creek hotlips magazines they work alot better than the factory ones. the factory ones i had didnt always want to keep feeding the ammo like they should but they could have a better design now since that was 4 or 5 years ago when i got mine.
What do you do with those empty nylon feed bags or dog & cat food bags? Can you say recycle? I use them for target practice for the youngsters. Take your empty shopping bags, stuff them into an empty nylon feed bag. You can also use old clothing or rags, (but put them in the bottom so it doesnt get top heavy). When your bag is full, fold the top over, punch some holes into the top and zip tie the top shut. Paint, draw, paste, or have the kids decorate their new target. Set it out 10 or 20...