you can use the rifles you have, but you really need to stay at closer ranges for them to work for you
22 50 yrd and less IMO and the 17 a 100 or less!
I still am not going to say the .357 is a good caliber, based on MY experience with it, its JUST a larger hole, and if things are closer, (i've seen what they can do to smaller things)
I wouldn't say there good on pelts if, your planning to save them!
maybe some better bullets and shot placement?? but I wouldnl;t call it a great coyote caliber, or one I would use is I had better options!
you might benefit from reading up on some ballistics's on the 22 and 17 and see where the energy levels are at at ranges!, and then base your distance on shots to take from some of that, and also consider when hunting and when stats on PAPER, are two different things and to maybe take shots slightly LESS than what things say they will do!
and these days when you have saved up a little $$ < there are a TON of great low costing options for rifles of better calibers!
.22 hornet .222-.223-.22-250 would be my 3 suggestions, with the .22-250 being my choice here, second would be the .223, , these two will also be found in the most rifles out there,makes/models!~
and last would be the .222, or 22 hornet , but less rifles will be chambered in these two calibers, making both less rifles offered in them, and ammo will be harder to find and just less options of it!
the .223 and .22-250 are both super popular and well tested and proven with LOTS of data and load options and have plenty of reach for hunting coyotes!