If you're taking the ACT without Writing, you should have time to send your September ACT scores to colleges even with the October 15th application due date. Multiple choice scores for the test are typically released about two weeks after the test. That means you'll get your full scores by the first week of October. It should take schools no more than two weeks to receive your scores electronically after you see them and send them out, which brings you to right around the application deadline.
If you're taking ACT with Writing, however, you might not be able to send your score in time for schools with October deadlines. Last year, after the first administration of the updated ACT Writing test, many students were upset to find that the grading of the Writing exam was taking longer than expected, mainly because ACT, Inc. will not send your scores to colleges until your entire test is graded. So even if your multiple choice test had been answered, if you'd taken the ACT with Writing and the essay wasn't scored yet, your scores wouldn't be sent out. It's to be hoped that with time, grading for the new ACT Writing test has gotten smoother and won't result in these kinds of delays...but ACT, Inc. does give the disclaimer that scores could take as long as 8 weeks to be available after the test date (which in most cases is after EA/ED deadlines).
As a precaution, I'd recommend contacting the admissions offices at colleges with the October deadlines to see what their specific policies are on receiving scores after the application deadline. Most schools don't have a problem receiving scores slightly later, but if you're feeling nervous about sending them so late you should definitely ask directly to give yourself some peace of mind.
As to sending your current scores, I'd say it's a good idea to send your 31 ACT score if it's not a huge financial hardship. If you improve on the September ACT, then just send those scores as well. If not, you're already done with your application.