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Auto comment: topic has been translated by Kane (original revision, translated revision, compare)
I first tried solving it with double hashing with primes 31 and 37. I got WA on Test 9. I changed the modulo, again WA on Test 9, so I realized it was because some hash was becoming zero along the way, and that's certainly not good when hashing...
My workaround was to make a list of primes and choose two randomly until all hashes are positive. I got AC with that. Here's the code -> C++ Double Hashing
I understand that you're asking specifically for the hashing solution, so you probably already know, but this problem can be solved simply by adding a copy of S to itself and then running KMP.
Here's the code for this solution: KMP Code (the hashing solution is above)