You cannot open a bank account in Thailand unless you already have a long term visa. So you cannot come to Thailand as a tourist with the plan of opening a bank account and then applying for a retirement visa, because you need a bank account to apply for the visa in Thailand.
Apply for the 90 day non-O visa online from the Thai embassy before you go to Thailand.
If it's not issued before you travel to Thailand, you'd be stamped in 60 days visa exempt. If it's then issued you would have to leave and return to use it.
You cannot apply for a non-OA visa from anywhere except your passport country.
And getting a non-O visa isn't possible without a Thai bank account, which you cannot get without a long term visa.
there are a few options if you'll actually run out a day early.
1) depending on your flight time, you can try checking in early and crossing through passport control before midnight
2) you can go 2-3 days before your extension expires and see if they'll accept an extension application that you don't qualify for. If they accept the application, they'll give you a 7-day "extension denied" stamp starting from the day of application.
3) overstay one day. There's no fine for one day overstay if leaving from an airport.
A tourist visa wouldn't benefit you. You would get 60 days when you entered and then that would be invalidated when you left 4 days later. Paying for a visa for 4 days would be pretty wasteful.
I'm a little confused. Which extension are you applying for? They will 100% not add an extra day for you, so you can forget about that option. But it's hard to say what options you might have because your post is confusing and doesn't include enough information like when you entered, when your entry stamp went until, what extension you are applying for and when it will go until.