apply for your regular 30 day extension first at your local immigration. Every entry is entitled to one 30 day extension on a tourist visa or visa exempt. After that is expired you need to leave the country and do the border re-entry process.
Although there’s no offical limits other than the 2 land borders, be wary about entering via the 2 major airports in particular if you have spent over 180 days in country in the past 12 months. The immigration officers at the Bangkok airports in particular are at their discretion pulling aside people who have been in country over 180 days as a regular tourist. It happened to me on 1st May and I needed approval by their superior to enter. If you are over that limit then you really need to look at other options to extending a visit ie making use of the 2 land border bounces (with a border run company) if not yet used, or applying for METV well in advance of a long planned trip.
If you are wanting another 60 days here and haven’t entered by land already twice this year on visa exempt then you could just do a border run using one of those border run company’s that get you out and back in one a day trip. It will get you 30 days with 30 more extension at immigration.
At least with e-visa you will still have your passport to enter visa-exempt (assume your country has visa exempt status). When I did mine last year I needed to post my passport with my application. With visa exempt you will need to ensure your airline will let you board without the visa or without an onward ticket for the 30 days that visa exempt provides. I have heard that some people did receive their visa late when already in Thailand. To active it you would just exit and re-enter the country with a printout of the visa PDF.
If you haven't got all the documentation ready to submit then you would be cutting it close to your departure date, but that would depend on your embassy's current processing time and whether they use the eVisa system as that may save you a few days.