It's a good problem to have! A rope or chain generally is ok for coho, but I find with Springs they are too long so I just have to carry it by the gills lest I drag it on the ground. When I know I am in for a long hike I will bring an external-frame pack, put the fish in a trash bag and put them head-down in the backpack - springs will have the tail sticking out of the top, nothing to be done about that. If your backpack carries your tackle, it could be hard to do this (I use a vest).
I've been looking for a suitably sized and shaped cooler backpack, but they don't seem to be tall enough.
Probably a soft-sided catch cooler would be a good option for you, as it could be folded and carried in your tackle backpack until you need it, then carried out with one hand full of fish - plus you can periodically set it down when it gets heavy without worrying about getting your fish dirty, etc.
Also, the old style vest with the large wrap-around back pocket has always been a popular way to carry out fish, but you better make sure it is in a good plastic bag first or your vest will really get slimed up.
As you can see, I've spent way too much time thinking about this.