Think about it: if your current

company run out of funding, will it keep you until end of October? Of course no, they have no money to pay your salary, and it will let you go immediately.

I believe that in your original offer letter, your employment is "at will", that is, both sides can terminate the employment at anytime. You may orally promised to work until the end of October, you should try your best to do so. But things always change, neither you or the company should absolutely honor a promise unless that is on paper and you signed it.