For longitudinal spin Seebeck effect (LSSE) devices, a multilayer structure comprising ferromagnetic and nonmagnetic layers is expected to improve their thermoelectric power. In this study, we developed the fabrication method for alternately stacked yttrium-iron-garnet (YIG)/Pt multilayer films on a gadolinium gallium garnet (GGG) (110) substrate, GGG/[YIG(49 nm)/Pt(4 nm)]$_n$ ($n =$ 1 - 5) based on room-temperature sputtering and $ex$-$situ$ post-annealing method and we evaluated their structural and LSSE properties. The fabricated [YIG/Pt]$_n$ samples show flat YIG/Pt interfaces and almost identical saturation magnetization $M_{rm s}$, although they contain polycrystalline YIG layers on Pt layers as well as single-crystalline YIG layers on GGG. In the samples, we observed clear LSSE signals and found that the LSSE thermoelectric power factor (PF) increases monotonically with increasing $n$; the PF of the [YIG/Pt]$_5$ sample is enhanced by a factor of $sim 28$ compared to that of [YIG/Pt]$_1$. This work may provide a guideline for developing future multilayerbased LSSE devices.