No. Your NPI is like a SSN.
This is incorrect. You should get another NPI for your part time practice as a "facility NPI" (i.e. your private practice is a technically a facility). Only bill using your facility NPI for private patients. Ideally, your personal NPI should also be separate for each facility, but if you already credentialed with the facility you worked at using your personal NPI, it's a bit too late...
Remember, insurance contracts goes with facility, not provider. Except Medicare. That one goes with provider. You are either in or out regardless of where you are. (Except you are not: you are par in, non-par in, or out).
That said, in your facility contract, especially you are full time, there might be a clause limiting your ability to practice outside of the facility. This is usually called "non-compete", though common parlance "non-compete clauses" refer to non-compete applications after termination. Nevertheless, in practice I've noticed that it's often very common for people to practice outside a facility ("moonlighting") even though their contracts said they shouldn't or all their clinical revenue should be routed to the facility. This is technically a breach of contract, and if the facility decides to sue you later to recover what they believe they are entitled to, it would be difficult to argue against in arbitration. However, as far as I know, this kind of lawsuit does not happen very frequently. Facilities don't typically care about the whatever tiny amount you generate through moonlighting and such arbitration could easily cost more than whatever is recovered, unless something very unusual happens (e.g. you started a competing facility on the side that "steals" patients).
Based on my experience, there are also LOTS of situations where the locum provider isn't technically credentialed (i.e. the individual NPI is not credentialed in-network with an insurance company and/or is opted out of Medicare). However, the facility (i.e. typically a hospital or community clinic) uses a different number to bill for reimbursement (i.e. their "medical director"). This is a legal gray area. Is the medical director personally involved in supervising? What kind of threshold is required for billing using a different provider number? Typically they are spelled out in group insurance contracts, but in practice insurance companies (even Medicare) RARELY file lawsuits or even audits for improprieties, unless such improprieties are egregious. Even when they do, audits often don't reveal much, as the provider may have long left the facility and responsibility for such improper billing rests in the facilities credentialing department. The facility might pay back whatever the audit reveals to be fined, but nobody is really personally held responsible. More likely scenario is the claims get rejected by the insurance company for "unable to locate provider in network", in which case the failed claim loops back to billing and gets reprocessed in some standard way. After repeated non-reimbursement the hospital eats the cost. Nobody cares. Locum's personal salary is typically not pegged to reimbursement, but rather, at worst, to billing. At best it's salaried or per hour. Audits of this type has been uncommon in mental health (compared to say gray zone billing for procedures), because the revenue is tiny and typically facilities lose money over them regardless of audits, and the response from the facility is typically--you want to fine me and shut me down, go right ahead. However, lately because of opioids and the rise of illicit rehabs and etc., some facilities are starting to become rather profitable, so perhaps one ought to prepare for more frequent audits of this type in the future.
These quirks of the American healthcare system also NP-related issues, as often group practices use credentialed physician NPIs to bill for NP visits for higher reimbursement. Technically, you are allowed to do it if the physician is the responsible person (like a resident supervised by an attending). In practice, the difference maybe the physician signs a note without knowing a lot except during team rounds.