@ HM2PAC,
Since you already have a .308 why don't give it a try first to see how you like coyote hunting. I would suggest using soft points over hollow points or FMJs. So what if you blow a big hole in them. There will always be more coyotes to go after if you catch the fever and decide on a dedicated predator rifle. And I don't care what anyone uses for a caliber / bullet / load selection eventually you will end up a coyote with either a big entrance or exit wound.
When I first started hunting coyotes around 40 years ago, I started with a .250-3000 or more commonly called a .250 Savage nowadays. It was a lever action model M99 Savage with peep sites. I was using the old silver tipped bullets. It wasn't exactly what I would call accurate, but it worked for taking down 'yotes until I could afford something better.
And for whatever it is worth I mainly use two different rifles to hunt coyotes lately. One is a plain vanilla Remington 700 VLS .22-.250 using out of the box Hornaday 50gr V-Max. The other is a custom built .243 with a Sako action, Lija 26" barrel with a 1:12 twist.