My advice is to freelance in some area that match your skills. It could be as a freelance writer, graphic designer, virtual assistant or any skill that you can pick up extra work from.
There are also part-time telecommuting jobs such as tutoring and customer service. For customer service, this time of the year is when they hired the most because of the holidays.
Get a subscription to FlexJobs for opportunities that aren't a scam or for free check on job or gig sites like Real Ways to Earn for all kinds of gig ideas, Problogger for writers, etc.
I second this. Freelance work based on you skill set can defintely be a good thing to do (or even spending some money on online courses to learn new skills). I used to work in retail management and learned/licensed in the financial services industry (its my main career and business now). But depending on what you decide to do, a side hustle has many tax benefits as a sole propriertor while still working your full time job, having the security to pay your bills and still be able to have family time.
First find something like to do and that fits your schedule. Over the years I have done lots of short term jobs for quick cash.
Including: working at big box nights/weekends, working mens better dressed at macys. Tax returns, I had an active cpa certificate for years and for about 10 years did tax returns on the side even I quit working in accounting full time. Once after I left accounting I found a small cpa firm that needed some one part time and would pay a premium for my Saturdays.
Are there really any penalties other than losing your job in the vast majority of situations?
In which case you have a second anyway... there's a lot of freedom in many dimensions in not really caring if one of the gigs goes away. The on call rotation thing isn't much of a concern: if things are breaking all the time pick a different company and other than that there generally isn't much overlap in my experience.
Ultimately if you get stuck you just pick one... and then go get another if you so choose once things have settled back down.
Will say as much as the income tax for little benefit to me is annoying in California, labor law is all in my favor. What are they going to do, fire me? Other states aren't as friendly towards labor admittedly... though in one little bit of amusement I was working at a vendor, and wound up on a project where said vendor's product was no bueno. Ah well I have plenty of experience fading into the background and made it through unscathed.
While in some cases there may be additional legal implications for doing so, in most cases termination is the worst that can happen. However, termination is a pretty big deal as I would imagine in most cases of people taking second jobs that one job is dominant/primary and has a much larger impact on them than the other. Your first job likely pays (significantly) more and has benefits you enjoy/need. Your second job is likely contrator-type work or part-time with limited/no benefits.
I suppose in the case of someone doing exclusively contractor-type work, jobs are jobs and they don't care how one feels about the other, but I imagine in most cases it's someone with a full-time career/staple job looking to supplement income with a second. In those cases, the penalty for job loss in the first by flirting with the second seems a poor decision to make.
Regarding things like on-call, it's rarely the case that the problem is things are breaking all the time. It's that your first job is paying you to be available should something break, and if you're doing a side-hustle as an Uber driver or fixing PCs at Geek Squad when that P1 call eventually does come in, what do you do? Walk out of Best Buy? Pull over with someone in your back seat? If you finish what you're doing before jumping on the on-call issue, you're sending the message to your employer that they should rethink why they're giving you a phone and paying you to do a job you aren't available to do. In the case of a profession in the medical field, the option to put off the response even 10-15 minutes can be life or death. I know a couple docs who would love to have a side-hustle (EDIT: second part-time job since it seems the definition of side hustle now doesn't necessarily mean second job) but simply can't for this reason alone.