What you're describing is called
remouillage in French, and means "rewetting." It makes a stronger, more gelatinous stock, but not a clearer one (which is what consommé is).
To make consommé, cold strained (and often, naturally jellied) stock is mixed with ground meat and vegetables, egg whites, and sometimes egg shells. This mixture is heated to a simmer and moved to the side of the burner so that only part of the pot is over direct heat. As the liquid bubbles, the solid ingredients coagulate into a "raft" that acts as a magnet for impurities such as barely visible bits of protein still in the stock.
Chicken consommé would use ground chicken for the ground meat; fish would use fish, and a white consommé, made from veal stock, would use beef or veal. Although frankly, I've never heard of a "white consommé." But then I don't know everything.
