What Mourinho needs is the right kind of players to play the way he wants. Giving him more time wouldnt change anything if he doesnt have the needed players.
Similar to Pep, he has a way in which he wants to set up his team, and thus only fits clubs that have or can get the type and quality of players needed. When they do have the players, they can produce a dominant side, but without the players that fit the needs, the output is less than desirable.
I think the transfers have been good so far (except for Doherty who seems to have an aversion for regular football). The key players needed going forward are a CM playmaker (help with ball retention and constructing attacks, instead of always losing it under the slightest press and inviting pressure back)and an elite CB (play a high backline and not need other players to be so defensive) . Those would likely be needed irrespective of the manager. So it would be better to get the needed players and see how Mourinho does, before going for a new manager.
With same players, its possible to have a new manager play better football, but dont see the results being much better. The squad is simply lacking some key players. Unless there is a generational talent youth player that you think the manager is best suited to developing, I dont see the miracle any manager is going work.
Of course, if the plan is to play possession football (like City and Barcelona), then Mourinho should be fired ASAP cos that would require a squad overhaul