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.
The record books, that are put out by Boone & Crocket and Pope & Young and where I live is Henry Kelsey they are a great way to research a state or a province for big game. This does take some time but the knowledge it gives you, can be very valuable. Not all hunters have their trophies recorded, as where I live after the Hanson Buck became the new world record typical Whitetail, we saw a flood of hunters and now that has happened, some hunters are shying away from the record books but...