That's a great opportunity to use an older scope off of a centerfire rifle and upgrade. One of my 10/22's has a 3-9 Simmons that I first had on my .243. My other bull barreled fun gun 10/22 has a 4-12 Simmons with AO. I'm not a fan of spending a lot on a rimfire scope.
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Was thinking about dropping a Nikon Buckmasters that I already have on my 10/22. Have a Marlin 22 mag with a Simmons 22 mag scope. It surprised me how well it shoots.
My marlin model 60 has a 3x9x40 swift my ruger 77/22 has a simmons actec, my remington 552 has a bushnell 3x9 tvscreen. I find on rimfires you dont have to have the best glass but have noticed how much i prefer a 40mm objective over 32mm.