This is why the earlier edition (2E IIRC) allowed different movement speeds as multiples of base speed and were better equipped than 5E in that respect. We have house-rules for running and sprinting as well. If you look at the calculations for a move + dash of 60 feet, that equates to 6.8 mph roughly, or a brisk jog (double-time).
Unless you have a class feature (e.g. Cunning Action), there is no way to "run" or triple your movement, or "sprint" which would be quadruple movement.
A "simple" house-rule would be something like this:
Run. When you take the dash action, you can use your bonus action to attempt to increase your movement again. You make a DC 5 Strength (Athletics) check. If you succeed, you gain additional movement equal to your speed, after applying any modifiers. The DC increases by 5 for each addition turn you run. If you spend one round without moving, the DC resets to 5.
Sprint. When you take the dash action, you can use your bonus action to attempt to increase your movement again. You make a DC 10 Strength (Athletics) check. If you succeed, you gain additional movement equal to twice your speed, after applying any modifiers. The DC increases by 5 for each addition turn you sprint. When you fail a check, you must spend a number of rounds equal to the number of rounds you spent sprinting using only your movement and no dash or run.
That is the basic idea. Tweak as desired.