Find shortest prefix to generate original string by overlapping
What you are looking for is called the quasiperiod of a string. If such a string has a quasiperiod of $|S|$ it is called superprimitive, and can not be covered by a substring.

A method for computing it in $O(n)$ time is given in "An On-Line String Superprimitivity Test" by Dany Breslauer. You might also be interested in "Of Periods, Quasiperiods, Repetitions and Covers" by Alberto Apostolico and Dany Breslauer.