Renumber values of a column in SQLite

Consider a table that has a position column. I like to renumber the position values so that each gets a unique value, starting with 1, and without gaps. The order does not matter.

I've tried this code:

UPDATE myTable SET position = (
    SELECT MAX(position)+1 FROM myTable
)

But that sets all values to the same value. Apparently, MAX(position) only gets evaluated once and not for every single updated row.

How do I make this work?

I've found a similar question where the solution is to use a separate table but I wonder if that's avoidable in my case.