cpsp,
TrevorP provides some good information regarding the considerations. Further to that great post, I would simply add that the amount of material can have a big impact on which technology is applied.
When the amount of material is very high and a continuous operation is warranted, a lot of people turn to things like the belt filter or the vacuum drum. Or even the horizontal decanter centrifuge. These devices tend to allow for more throughput but at the expense (generally speaking) of higher moisture in the dewatered cake.
In any event, the amount of solids is a very important consideration. If you are looking for alternated because the recessed press can't keep up, then you already know this. But if you shift gears, keep in mind that the other options generally aren't as effective at drying, especially if you have a high-pressure press, a.k.a. a membrane-type.