Score reporting: the SAT and the SAT with essay

The schools I apply to all require the SAT or the SAT with essay.
If I take an SAT test after I already submitted in an Early Action application (say, in December) , do I need to send in another SAT score report to the school just for the newest test? I remember reading about how I should update transcripts and the like in general, but am I required to update information from retaking standardized tests?

Note: I already have a SAT score I’m satisfied with, I just need the essay portion for a few schools.

Also, the language on Caltech’s website seems to say that they don’t require nor consider the essay scores if I try to send them in, so there seems to be no point in doing so. If I’m interpreting the language right, will they ignore the essay score even if I hypothetically get a 6/24? Will they treat two applicants the same if they theoretically have the same stats, except one has a 24/24 and the other a 6/24 on the essay? If they don’t, can I withhold my SAT with essay score if I don’t like the score?

If they say they don’t consider it, they won’t look at it. You probably should’ve taken the SAT with essay earlier, as you might not have time to submit them before school deadlines for scores. But you should send in all your score reports to all your schools.