I'm betting you will see an improvement with Magnum primers.
I'm not telling anyone to exceed published loads as a rule but like I said above, the load data for HS-6 is anemic at best. Look around and look at the .357 Magnum load data on the
Hodgdon Load Data Site for lead 158gr bullets and you will see a much higher charge listed than in the .38 Special but less pressure from .7gr more powder. Also, look at some older Winchester W540 data for real results. They are both the same powders.
Hodgdon lists a charge of 7.0gr with pressures of only 15,500 CUP in .357 Magnum brass using a Magnum primer. Granted the Magnum case is 1/8" longer but it's hard to believe you can increase the charge weight that much and have such low pressure from only 1/8" more capacity.
This is only my opinion, I'm not telling anyone to do something they feel is unsafe. I'm only thinking out loud and extrapolating data.
With a .38 Special +P pressure ceiling of 20,000 PSI which is something like 22,000 CUP I'm sure 7.0gr is safe in a .38 Special case.
Again, just my opinion, I'm not telling anyone to do what I do.