On your 1099 sub-contract with a prime, you'll likely find a non-solicitation cause that basically says you can't recruit the prime's employees and vice versa. Usually it's more nuanced in that you can still hire someone if they randomly apply on your company website, but you can't go up to Joe from the prime and try to get him to work for you.
That's fine because it's mostly not relevant if you want to stay solo.
BUT, where it may come into play is if you try to get a 1099 gig with a company that has partnered with your current employer.
For example, if you work for a sub-contractor and you want to cut them out and work directly for the prime, the prime program manager may be a little squeamish about bringing you on due to the non-solicitation agreement and the relationship with the sub (your employer).
I haven't directly encountered this so this may be more hypothetical than real, but if you do get into a conversation with a prime that cites the non-soliciation agreement as a reason, then say something like this:
"I totally understand your concern.
You don't want to risk violating your contract terms with my company. However, because I'm approaching you first and I'm specifically looking to 1099 rather than become an employee, my understanding is we wouldn't be violating the non-solicitation agreement.
Would you disagree? If not I'd love to keep the conversation going."
A successful 1099 sub-contractor will learn how to push through minor administrative obstacles, like pesky non-solicitation agreements.