Jordon,
Sunrig pretty much covered the issue.
I'd suggest a few additional considerations.
For discussion's sake, let's say you work for XYZ Airlines:
Do they have routes to the city where you want to commute? With what frequency ? Is it seasonal or been in place forever ? It's always easier to ride on your own airline.
Do you have seniority to manipulate your schedule to get long stretches of time off to make the commuting hassle worth the effort ?
Assuming you'd be using your travel privileges as a new pilot, what is the boarding priority as a pass rider...date of hire ? Some other protocol ?
Do you have access to the cockpit jump seat on XYZ Airlines ?
Do you have access to the cockpit jump seat on other airlines ?
Can you use your pass privileges on another airline or get a discounted ticket on another airline ?
Sunrig is right that people commute from all over the world but they have to have some seniority horsepower to make it work.
Atlas has a system worth investigating but they aren't a "major carrier" if you're referring to the likes to Delta, American, United, JetBlue, etc. But an ACMI carrier may be exactly what you need.
Good luck. You've got your work cut out for you.