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Frustrated magnets with degenerate ground states exhibit exotic ground statesand rich phase structures when perturbations and/or thermal fluctuations liftthe degeneracy. In two-dimensional models with short-range interactions,continuous symmetries cannot spontaneously break at finite temperatures,leading to the suppression of conventional magnetic long-range ordering (LRO).In this paper, we numerically study the classical $J_1$-$J_2$ $XY$antiferromagnet on the kagome lattice as a prototype model of such frustratedmagnets, where $J_2$ denotes the next-nearest-neighbor exchange interaction. Wemap out the $J_2$-$T$ phase diagram of this model employing extensive classicalMonte Carlo (MC) simulations. The obtained phase diagram featuresBerezinskii-Kosterlitz-Thouless (BKT) transitions of $q=0$,$\sqrt{3}\times\sqrt{3}$ magnetic orders, and octupole orders, in addition tofinite-temperature phase transitions of both ferrochiral and antiferrochirallong-range orders. Additionally, we find a non-trivial first-order transitionfor antiferromagnetic $J_2/J_1 < 0$. The origin of this transition is discussedin the context of non-local loop structures present in local $120^\circ$ spinstructures.

Monte Carlo study on low-temperature phase diagrams of the $J_1$-$J_2$ classical $XY$ kagome antiferromagnet