I have a BSA Contender 4-16 on my M&P 15-22 and a Simmons 3-9 with 22 Turrets on my 10/22. Both spend time in the truck bouncing around the ranch to keep the prairie dogs and ground squirrels in check. For an inexpensive option, I am really impressed with both scopes.
The BSA appears to have been discontinued:
Shop BSA Optics Contender Series Rifle Scopes 4-16 X40 mm | 4.3 Star Rating on 22 Reviews for BSA Optics Contender Series Rifle Scopes 4-16 X40 mm + Free Shipping over...
Nikon Pro Staff Rimfire
I have EFR on Weihrauch break barrel air rifle and 4x on 10/22 but they have variable models for rimfire also.
Great glass for the money.
Really enjoy the Bushnell AR Optics 2-7x32 on my 22LR. It has side parallax that will go all the way down to 10 yards IIRC, and a reticle meant for 22 LR.
A couple years ago Cabela's put their caliber specific rimfire scopes on sale for $50. They're all 3-9x40's. I bought one for a .22LR and liked it so much that I bought three more for that price. I have two .17HMR's with them, a Ruger 10/22 with one, and a Henry lever-action .22LR with one. Every rifle has been an absolute small critter killing machine. Pretty impressed with them. They're normally $99.99.