Okay Greg, let's see here.
It's legal to hunt coyotes in Wa. State if you have a valid hunting license.
It's legal to hunt coyotes using a centerfire cartridge of .224 or less during deer and elk seasons without the need of a big game tag, however, you must wear hunter orange during modern firearm deer and elk seasons.
It's legal to hunt coyotes during deer and elk modern firearm seasons with centerfire cartridge 6mm or larger or containing slugs or buckshot for big game IF you have a (any) large game tag in your possession. That means any tag for deer, elk, bear or cougar must be on your person during the time of hunt.
During archery season, you may only hunt big game with a bow that is legal to use during archery season, no cross bows and some other BS things they decided to include. A person SHALL NOT carry any type of firearm or have a firearm in or on your person while achery hunting deer or elk. My guess is they have superceded the 2nd amendment right to carry a concealed handgun during archery season if you are in the act of hunting with a bow even if you are a licensed concealed carry permit holder. Don't shoot a deer or elk with a bow then go finish it off with a handgun, that is a big no no.
During modern firearm season you may hunt with firearm, bow or muzzle loader providing you have the modern firearm tag for that big game species.
The State of Washington allows the open carry of any legal firearm upon any legal person that has not been convicted of a felony crime or convicted of any domestic violence laws causing or making that person unlawful to carry such firearm. And there is a few other things in there somewhere that one should read up on if they have any doubts as to their being a legal and lawful citizen. A legal citizen may carry a firearm into the wilderness, woods, bushes out back and a few other places as long as there is no intent to intimidate or threaten another party or property with the lawful act of carrying such a piece. So get a big gun and wear it proudly. Just don't make someone else feel uncomfortable around you, in other words don't go into Seattle with a 44 mag strapped on yer hip and saying things like....GO AHEAD PUNK, MAKE MY DAY
Carrying a concealed pistol upon your person requires the person to hold a valid Washington State concealed pistol permit.
http://wdfw.wa.gov/wlm/game/hunter/huntregs2005.pdf