It seems that formula (5.7) of Chapter XIV of the 1st volume of Feller may be used to obtain the full distribution of $C_n$ (as distribution of sum of two independent r.v.'s with explicit distributions). Indeed, to cover the path you must visit both extremes. First, use that formula to obtain the distribution of the hitting time of $\{1,n\}$. Once you hit one of the extremes, you have to go to the other one, and the distribution of this time is the same as the distribution of hitting time of $\{-(n-1),n-1\}$ starting at $0$.

It seems that formula (5.7) of Chapter XIV of the 1st volume of Feller may be used to obtain the full distribution of $C_n$ (as distribution of sum of two independent r.v.'s with explicit distributions). Indeed, to cover the path you must visit both extremes. First, use that formula to obtain the distribution of the hitting time of $\{1,n\}$. Once you hit one of the extremes, you have to go to the other one, and the distribution of this time is the same as the distribution of hitting time of $\{-(n-1),n-1\}$ starting at $0$.

Also, see formula (4.11) of the same chapter for the expression for the generating function of the hitting time; this may help obtaining the moments of the cover time.