Making soy milk: best way to separate okara?
When straining things that clog up the screen, I tend to use a spoon or spatula to move the stuff around, so you can get expose un-obstructed parts of the screen.

Of course, you might end up pushing some of the pulp through the process, but it's generally minimal.

You can then either dump out the pulp as you go (possibly into muslin or cheese cloth, let it cool, then give that part a squeeze), or use the back of a large spoon or a rubber scraper to push the pulp against the strainer to get out some extra liquid. (or use an oven mitt, covered in a plastic bag ... it's messy, but it works).

Of course, if you have a cheap strainer, too much force on it can cause the whole screen to pop out, splashing you and making a quite large mess.  (I've since made sure that my strainers weren't basically a screen just held in by pressure to a loop of metal that attached to the handle, but are actually one integral part).

...

If you're looking at buying new equipment, I'd look into getting a chinoise with the proper sized stick (not sure what it's called), which you can use to basically compress all of the pulp into the bottom as you're working.