Yes, you can absolutely do that, or send the other portion to a conventional IRA of your choosing.
The self-directed IRA is at most levels an IRA like any other. It can accept contributions and rollovers from other plans in the same fashion as any other IRA, either during the initial setup or in the future.
The one thing I would be sure to check on is whether your new 401(k) would let you roll out any of the funds you are rolling in. Most will allow that, but some may not. You may not want to have that capital trapped in that plan until such time as you change jobs or reach age 59 1/2.
@Brian Eastman gave a spot on answer, from my perspective as a account holder side of things.
My own thought is why would you put ANY of it into a Company 401K instead of a Traditional or Self Directed IRA? IF they allowed you to 'roll it back out' if you wanted then it might be OK, but even then, I would think the fees in that program would far outweigh having an IRA, and you would have more flexibility too.
A friend of mine rolled his into a SDIRA, and DOES do his new companies 401K with NEW contributions since they match those at up to 6% of his salary. A side benefit of doing ANY level of new contributions is that you then get the 'company recommendations' on funds which you can mirror with your IRA if you desire, at a lower cost.
Sometimes 401ks have ambiguous rules @Brian Eastman discussed-make sure you understand them. Some will limit you to one move. Try to do a direct rollover “custodian to custodian “so you don’t pay 20% income taxes. Try not to have the old 401k company make the check out to you personally but instead directly to the new plan(s).
I'd put it all in the SD IRA. I use CamaPlan and have been very happy with them. If you know enough to have an SD IRA, then you will likely generate a greater yield with that than with the company 401(k).
Certainly, you can accomplish both. You may later also transfer retirement funds that you may have transferred to the new employer plan if that new 401k plan allows for outgoing transfers of amounts rolled into the plan.
Also, while the full-time employer plan is protected from creditors under ERISA, not all IRAs are protected from creditors since creditor protection for IRAs is based on state by state statutory protection.
It appears, however, that Louisiana for the most part offers good protection to IRAs, per the following.
100% - La. Rev. Stat. Ann. Sec 20:33(1) - for funds deposited at least 1 year prior and 13:3881 (D)
You can roll all or part of that old 401(k) to an IRA of your choosing, including a self-directed IRA that provides broader investment choices.
IRA plans do not offer a participant loan feature as 401(k) plans do, however.
If you are self-employed, you could potentially establish your own Solo 401(k) and then have access to the loan feature.
If you will be joining another W-2 employer you could also potentially roll some of your current 401(k) into that plan such that you would have some capital from which to borrow.
If you have an outstanding loan on your prior employer 401(k) that may need to be paid off promptly if you have terminated service.
Unlike a 401k plan, the IRA rules do not allow for participant loans.
Unfortunately, you cannot borrow from an IRA. That would be one of the benefits a 401k plan can offer that IRAs do not. If you are eligible for a Solo 401k, you may be able to get the best of both worlds, but it does require some self-employment activity.