Stamina gradually degrades over time for all players.
Hiring a nutritionist can slow that decay (or stop it at level 6 and allow for very slow growth at level 7), but is generally considered a poor investment, especially for teams that aren't fully established and swimming in extra cash. Having a trainer with a fitness specialty will slow the decay as well.
The only other way to raise stamina is through the Team Training > Stamina training option. Which will give each player approximately 0.67 of a level in stamina increase (subject to a hard cap at 10.0) at the expense of a training week, and like all Team Training methods it does not depend on how many minutes the player played during the week or at what position(s).