I have two Simmons 3 -9 x 32 .22 Mag with A.O. scopes that have worked very well on my 10-22's for years. They're hard to beat. Cost is about $45 to $65 depending on finish and sales. Also, do yourself a favor and spend another $20 for some good Weaver rings. The rings that come with many scopes aren't that good.
A few years ago I needed an inexpensive scope for my old bolt action .22 rifle and grabbed a CenterPoint 3-9X32 .22 Rimfire scope at Walmart for something like $54. I have been super impressed with the quality and clarity of this scope especilly considering the price. It has held zero perfectly and I really like the, mil-dot retical it has. I zero at 50 yards and then use a two-dot holdover at 100 yards to be on target.