LEHB, not to hijack, but side pressure definitely gets them in quicker. A fish swims with side to side motion so if using side pressure at least every other stroke of the tail brings the fish closer to you. I also recommend to put more pressure when the fish is facing away from you than when he is facing you, so that you are pulling the hook back into the mouth rather than straight out the front.
A. H. E. Wood states in “Greased Line Fishing for Salmon [and Steelhead] by Jock Scott Introduction by Bill Mc Millan” “Down and to the bank.” Fish headed down stream turn upstream. very logical, when the rod tip is up then lowered horizontal the river bank to point downstream the fly line gets a bit of slack, a downstream curve is formed, and the tug from the line on the fish changes direction. a fish will try to pull away from the direction of the teather so then the fish thinks he needs to go updtream to escape and as tigfly has noted the hook pulls into the mouth.