Japt, 7 bytes
ÈsY}f+U
ÈsY}f+U :Implicit input of string U
È :Function taking a string and a 0-based iteration index Y as arguments
sY : Slice the string from the Yth character
} :End function
f :Get the first string that returns falsey (empty string)
+U : Where each string is U appended to the current iteration index