How can I obtain an additional week on my Thai visa extension by providing proof of flights?

It's actually not an extension but an "application for extension not approved" stamp that gives you 7 days to get out of the country from the date it's issued (and it starts the day you apply for it). It cost the same as a 'regular' extension 1900baht

I don't know that you need to prove you have a flight out to get it issued but couldn't hurt.

Go to your immigration office and apply for an extension after you already got the one 30 day extension, it will be denied and they will sell you a 7 day get out of dodge stamp