Here's one that claims to produce 75% less less sound than a standard velocity load. Seems like a
very good start on suppresion, right out of the gate!

Be aware that it likely won't operate your 10/22's action (they tell you that, right on the label). This one is the speed of CCI's CB Caps, but with a 40-grain bullet, vs. the CB's 29 grain bullet.
https://www.cci-ammunition.com/rimfire/cci/quiet-22/6-960.html———
I hear great things about the segmented rounds, as well- bullet is designed to split into three inside your quarry. Rabbits aren't necessarily very hard to kill in the first place, but these types of loads are moving pretty slowly. Three wound channels might be better than one! Here's a "segmented" version of the above "Quiet" loading.
This is probably the first one I'd try, balancing minimal noise with the segmented bullet. (Don't know why the link isn't posting a pretty picture…)
https://www.cci-ammunition.com/rimfire/cci/quiet-22-segmented-hp/6-970.html[ snip]
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Now if they'd just load the segmented bullet into a "Quiet" load that would cycle the semi-auto… then we might be in business!