What can I do if I need my passport as soon as possible, but there is no available slot online for an appointment this month?

You may try checking from time to time to see if an appointment slot becomes available (which can be highly unlikely). You may also try going to a travel agency near you to facilitate in checking your requirements, setting an appointment, and providing other instructions. There is an additional fee for handling sometimes, depending on the travel agency. As of the moment, DFA Aseana has a separate queue for travel agency applicants which could move quite fast due to the agency ensuring that you have the necessary documents and payments already settled. You'd just need to go to the encoding section, confirm the information, and then wait for your passport to be processed.