So most other people have taken the "yes it's possible" answer off the table.
The bigger question is how are you going to actually do it.
Some hospital systems may actually have Gen Peds urgent care or hospitalist shifts available - which would likely be the easiest as you wouldn't have to travel and your program director is likely aware that their fellows have this opportunity. Credentialing should be easier, though depending on set up you might run into a small bit of wonkiness with your paychecks (hopefully not if this is a common thing for people to do).
If that's not an option, you'll need to find a company that will scout opportunities for you. Google "locum tenens general pediatrics" or "locum tenens job boards" and you'll come across a ton of different companies. When I was coming out of fellowship, I actually found a website (that I don't remember the name of now) where I put my name/info in and received calls from easily a dozen different locums companies with opportunities - some actually called me within hours of posting my info. You should do your due diligence though on companies though to figure out who is well regarded. (Personally, my favorite was CompHealth, but there are lots of options). You'll find that some of the recruiters just feel "off" and others don't follow up in a way that is useful.
Don't let recruiters bully you into positions that don't fit for you - whether it's location, hours, or payment rates. If there is some aspect that you don't want to do, be firm and clear (for example, as a PICU attending, when I was doing locums, I refused to do newborn nursery jobs, one company lied to me telling me it was a "pediatric hospitalist" spot and I ended up in a gig seeing nothing older than 72 hours). Whether it's supervising residents, taking calls, or certain volume, remember that they need you more than you need them. Also, if they come in below your expected pay rate, ask them to go back to the site to see if there is any wiggle room in the dollar amount.
If you are willing to travel out of state, make the locums companies handle all of your fees - whether it's an additional DEA number, state licenses, etc - you should not spend money on things to make you presentable. The only instances I can think where that would be worthwhile would be if you have an interest in doing work in a state that is notoriously slow for licensing (eg California or Texas, among others). Like if your parents live in Texas and you knew that you could just head home for a spot of time and do some sidework, then having that state license already would increase your opportunities. Many states that have a robust locums market actually have mechanisms in place to fast track licensing (Indiana is incredibly fast from personal experience).
Lastly, 100% do not tried to sidestep your fellowship director. Nothing good will come of it if your fellowship director doesn't feel like they can trust you. If they refuse initially, speak with them directly and lay out why you need this opportunity and show how it won't affect your work as a fellow.
