@PaulSiegel: Thank you very much for your thorough reply, I really appreciate it! I will look at the paper that you mentioned, but I already got what I need from what you wrote.

@PaulSiegel: Thanks for your reply. Do you know a counter-example when the covering map is required to be smooth (not just continuous)? This is what I had intended to ask originally, as I didn't expect any subtlety involving exotic spheres.

@LeeMosher: You are right. I thought about this but decided to edit the question all the same because Paul had replied almost immediately after the question had been posted, and because the edit consisted of 1 word only. I have followed your suggestion to revert to the original question and accept his answer which is perfectly adequate for the original question. I will however not post a new question as it is too similar to this one. I will leave the question that I had intended as a comment (see below). I apologize to everyone for my mistake in the original statement.

@PaulBryan I had some vague knowledge of results on geometric flows such as the Gage-Hamilton flow and thought that something along these lines could work. Thank you very much for your help. I will take a look at the paper that you mentioned.

@BenMcKay I had thought about this, but could not see geometrically that it would work and wanted to avoid going through the calculations if possible, which should be fairly complicated. It is for instance not true that linear contraction using stereographic projection (instead of the exponential map) will work. Thank you for your help!