In this paper we define new spectral properties $(h)$, $(gh)$, $(ah)$ and $(gah)$ as a continuation of [H. Zariouh, Property (gz) for bounded linear operators, Mat. Vesnik, 65 (2013), 94-103], which are variants to the properties $(z)$, $(gz)$ and $(az)$ introduced in the mentioned paper. The purpose of the paper is to study the relationship between these properties and other Weyl-type theorems.