The integer Carathéodory rank of a pointed rational cone $C$ is the smallest number $k$ such that every integer vector contained in $C$ is an integral non-negative combination of at most $k$ Hilbert basis elements. We investigate the integer Carathéodory rank of simplicial cones with respect to their multiplicity, i.e., the determinant of the integral generators of the cone. One of the main results states that simplicial cones with multiplicity bounded by five have the integral Carathéodory property, that is, the integer Carathéodory rank equals the dimension. Furthermore, we present a novel upper bound on the integer Carathéodory rank which depends on the dimension and the multiplicity. This bound improves upon the best known upper bound on the integer Carathéodory rank if the dimension exceeds the multiplicity. At last, we present special cones which have the integral Carathéodory property such as certain dual cones of Gorenstein cones.