We introduce minislotted threshold ALOHA (MiSTA), a slotted ALOHA modification designed to minimize the network-wide time average Age of Information (AoI). In MiSTA, sources whose ages are below a certain threshold stay silent. A node with the age above the threshold becomes active in the next time frame with a certain probability. The active node first transmits a short control sequence in a minislot ahead of actual data transmission, and if collision is sensed, it backs off with a certain probability. We derive the steady-state distribution of the number of active sources and analyze its limiting behavior. We show that MiSTA probabilistically converges to a "thinned" slotted ALOHA, where the number of active users at steady state adjusts to optimize age. With an optimal selection of parameters, MiSTA achieves an AoI scaling with the number of sources, n, as 0.9641n, which is an improvement over the threshold ALOHA policy proposed earlier (for which the lowest possible scaling is 1.4169n). While achieving this reduction in age, MiSTA also increases the theoretically achievable throughput to approximately 53 %, from the 37% achievable by threshold ALOHA and regular slotted ALOHA.