Someone else mentioned using a water jet. We did this on one troller I worked on. You head the fish before opening the belly, and there are two arteries exposed, near the spine. Then you insert the water jet into one of them and the water works through the entire circulatory system in seconds. It is super cool to see it in action. The jet was something like a water pick attached to a piece of aquarium tubing then connected to the deck hose. Frankly, a bit messy and a bit more work. I'm not sure it is any better than just bleeding as soon as bonked. But it sure does clean the blood out.
To answer someone else's question, slime is good. The hold on a hali boat is a total slime fest, but the fish is first class (if properly dressed and belly iced). And all halis have "nuts", boys and girls. If you're icing them, then rip em out! Scrape all the "sweet meat" too.
And for those going after tuna, brain spike and bleed right away. We'd bleed ours in a tub of water, so that once the blood was pumped out water would also pump through... not sure if it worked that way, but that's the theory.
One exception to all this is sockeye. If you are going to be freezing them right away (not storing on ice for days), you can bleed them and freeze them on the round, guts and all. They keep amazingly well this way, for a looong time. Only sockeye though, the rest all have too much stomach acid.