If you are looking at doing 3 miles faster you need to up your overall mileage in the week. I would suggest getting anywhere from 4-6 miles in per day and do that at whatever pace makes you comfortable. Start off with not timing yourself, just finish. Then increase your time, but be happy with consistency off the back.
You need to build endurance. Anything past a mile to me is more of an endurance race to me. A mile and below(especially a 1/2mile and below) I call a sprint. Once you can successfully achieve 4-6 miles daily you will build the endurance needed to pick up your speed for a shorter race.
Your analogy might be a tad off base for running, but i saw what you were trying to achieve, i just asked to make sure. Unlike weight lifting, you can not push the shortest distance possible for the longest amount of time in hopes of gaining distance endurance.
I ran the 1/2 mile, mile, and 2 mile all in the same track event, every event. The only way to have endurance for all three was to put miles under my belt. Maybe 40-50 miles a week. If you want to build speed, interval training is a great choice too. I would suggest mixing it up with 1/4 mile intervals or shorter. When we were at the point of peaking we used to do mile intervals and that was killer. Trying running a 4:55, two minute break, then running a 4:57. two minute break, 4:56, two minute break, then a 5:00. Do something similar one day a week with shorter distance until you can build to something like the above. Maybe 100m, 30 seconds break, 100m, 30 second break...
Then just pick up the distances slowly. ENDURANCE, MILES. This is the only way to lower a time in the three mile range.
great info man. I do appreciate that. I will consider everything and check into it further. But you did answer my question