Can I use my 60-day visa after entering Thailand visa exempt on my first entry?

This is exactly what I did, so yes it is possible!

My Tourist Visa was not approved in time for my flight into Thailand, so I had to use a 45 day stamp on arrival, which I then extended for the 30 days.

During this time I have had my now approved and unused Tourist Visa, which I successfully used 3 weeks ago when doing a Malaysia land border run.

Edit: But now I think about it, maybe because your Tourist Visa is already approved at the time of entry (mine wasn't) then your situation is different

As a rule it will totally be up to the immigration officer as far as IF you can get in visa exempt and then activate the tourist visa when you come back into thailand by land.

If it's an eVisa <- that you got online by the PDF email, there have been reports of people who didn't present it, entered visa exempt and then when they went to use the eVisa it was already invalid.

If it's a sticker visa that you got at a thai consulate, all you can do is ask the officer, but keep in mind they are under no obligation to let you in visa exempt and 'save' that tourist visa for another entry

Worst case IF you have to enter on the tourist visa, go to the immigration office where you will stay for the 4 days and BUY a single re-entry permit for 1000baht, that will let you get back in on the tourist visa you entered on and get the remaining time left on it

Is it in your passport?