Expressive temporal reasoning formalisms are essential for AI. One family of such formalisms consists of disjunctive extensions of the simple temporal problem (STP). Such extensions are well studied in the literature and they have many important applications. It is known that deciding satisfiability of disjunctive STPs is NP-hard, while the fine-grained complexity of such problems is virtually unexplored. We present novel algorithms that exploit structural properties of the solution space and prove, assuming the Exponential-Time Hypothesis, that their worst-case time complexity is close to optimal. Among other things, we make progress towards resolving a long-open question concerning whether Allens interval algebra can be solved in single-exponential time, by giving a 2(O(n log log n)) algorithm for the special case of unit-length intervals.