Let $p\ge 5$ be a prime and let $n$ be a natural number. In this article we describe the irreducible constituents of the induced characters $\varphi \uparrow Sn$ for arbitrary linear characters $\varphi$ of a Sylow $p$-subgroup of the symmetric group $Sn$, generalising earlier results of the authors. By doing so, we introduce Sylow branching coefficients for symmetric groups.