For inexpensive 22, a semi auto is probably a better deal overall than a bolt action. A locked action (bolt) is not necessary for .22 rimfire, and it costs more to make.
A .22 semi auto is better than a bolt action, if you want a semi auto, and, if it works. And, when talking about "budget grade .22s" bolt guns are cheaper to make than semi autos. Often considerably. They are simpler, with fewer parts, and often even cheap .22 bolt guns will have better triggers than semis, and if not, are easier to improve, if you desire to do so.
OP, other than personal preference, why does it have to be new??
There are millions upon millions of used .22s in serviceable condition all over the country, generally priced less than the cheapest new production anything.
The OP said he wanted something cheap, to shoot the ammo his 10/22 didn't like. He didn't specific a repeater, or action type.
This makes me think of the rifle I learned to shoot with, my Mother's rifle, a JC Higgins single shot bolt action, which I learned to shoot as a boy of 10 (when I was strong enough to cock it on my own
). Accurate enough to center punch poker chips to 50yds and make head shots on small game back when high grade .22LR was $0.39 a box.