It has been proposed [Ginelli et al., Phys. Rev. E 71, 026121 (2005)] that, unlike the short-range contact process, the long-range counterpart may lead to the existence of a discontinuous phase transition in one dimension. Aiming to explore such a link, here we investigate thoroughly a family of long-range contact processes. They are introduced through the transition rate 1+aℓ(-σ), where ℓ is the length of inactive islands surrounding particles. In the former approach we reconsider the original model (called the σ-contact process) by considering distinct mechanisms of weakening the long-range interaction toward the short-range limit. In addition, we study the effect of different rules, including creation and annihilation by clusters of particles and distinct versions with infinitely many absorbing states. Our results show that for all examples presenting a single absorbing state, a discontinuous transition is possible for small σ. On the other hand, the presence of infinite absorbing states leads to a distinct scenario depending on the interactions at the perimeter of inactive sites.