Is there any general method to calculate the universal cover of a given topological space $X$? I am currently taking a course in algebraic topology and I have to calculate the universal cover of a lot of spaces (I'll call them $X$ from here on). So far I know some tricks to do it: consider $X'$ simply connected and a group $\Gamma$ which acts proper and discontinuously on $X'$ such that $X \cong X'/\Gamma $ (I calculated the universal covers of the Möbius band, the Klein bottle and the torus with this), try to imagine the covering space knowing that if the spaces have "nice" properties $\#F_p = \# \pi_1(X)$ (I calculated the universal cover of $S^2$ with a loop attached with this), $\dots$
I feel that these methods rely a lot on intuition and I would like to use a general method if possible to avoid having trouble when doing the computation due to my lack of "vision". Is there any trick or general method which I can use to do this calculations? 

Here's an 'algorithm' that may be helpful in the special case when $X$ is a wedge of two topological spaces: http://www.math3ma.com/mathema/2015/12/17/a-recipe-for-the-universal-cover-of-xy
