The first possibility is to rerun the local search algorithm with a different starting solution. This will generally lead to different local optimal solutions. This is one of the big advantage of the random heuristic: it is very easy to generate different starts. But many other heuristics have the ability to generate different solutions. Consider nearest neighbor for the TSP: it is completely arbitrary at which city the tour is started. But different starting solutions lead to different tours.
Another area where we can generate different local optimal solutions is in the choice of exchange. Note that there may be many exchanges that improve the solution. A different choice of which exchange to accept may lead to different local optimal solutions.
The main advantage of this approach is the ease of implementation and the quickness. Unfortunately, even iterating might not provide a good enough solution.