If you can't find any .222 diameter bullets, then Sierra used to make and probably still does, a .223 diameter bullet for the .22 Hornet. I am sure you can use these with no problems. 8)
I checked and they still do make the .223 diameter bullets.
https://www.sierrabullets.com/products/bullets/rifle.cfm/caliberID/2
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I have the Hornady jet bullets. And no brass. $15 box, plus shipping I have 5 boxes. I can trade if you have something I can use. I also have the steel inserts for shooting 22 lr.