Minimal numbers of linear constituents in Sylow restrictions for symmetric groups

Let $p$ be any prime. We determine precisely those irreducible characters of symmetric groups which contain at most $p$ distinct linear constituents in their restriction to a Sylow $p$-subgroup, answering a question of Giannelli and Navarro. Moreover, we identify all of the linear constituents of such characters, and in the case $p=2$ explicitly calculate a new class of Sylow branching coefficients for symmetric groups indexed by so-called almost hook partitions.