A guide friend of mine showed me a neat trick when cleaning fish.
He cleans his fish on a cleaning table with a hose handy and he uses the end of the hose (no connector) to push the blood out of the stomach viens.
Once you take the kidneys out (bloody organ along the spine) you can push the blood out there.
Makes for a very clean fish.
Obviously, you don't have a hose at the side of the river but you can use another similarly shaped object to push the blood out.
I typically cut the gills right after bonking and leave the fish in the water to prevent coagulation of the blood (keeps the blood flowing for a few minutes).
I see alot of gills strewn about the bar when the fishing is heavy.