The aim of this paper is to lay a foundation for providing a soft algebraic tool in considering many problems that contains uncertainties. In order to provide these soft algebraic structures, we introduce the concepts of $SI$-$h$-bi-ideals and $SI$-$h$-quasi-ideals of hemirings. The relationships between these kinds of soft intersection $h$-ideals are established. Finally, some characterizations of $h$-hemiregular, $h$-intra-hemiregular and $h$-quasi-hemiregular hemirings are investigated by these kinds of soft intersection $h$-ideals.