An n-color composition is one in which a part of size m can come in m colors (denoted by subscripts). Let C(ν) denote the set of n-color compositions of the positive integer ν. In this paper, we consider further modular restrictions on the subscripts of the parts within members of C(ν). We first count members of C(ν) in which all parts have subscripts of the form ℓa+b, where b and ℓ are fixed and a ≥ 0 is arbitrary. Generating function and explicit formulas are found for general b and ℓ which extend earlier results when ℓ = 2 and b ≤ 3. We study the case ℓ = b-1 in further detail and find that the corresponding subset of C(ν) is in bijection with various classes of compositions. Finally, we consider two related problems: one where the subscript restriction applies only to parts within a given modular class and another where the subscript of a part belongs to the same modular class mod ℓ as the part where ℓ is fixed.