Is it possible to combine health insurance such as Social Security along with a private health coverage to get an upgrade in care?

I'm not sure what you mean by upgrade in care since social security already covers a semi-private room.

It is certainly possible to have both private insurance and social security. The main advantage would be more range of choice since social security will only pay for care at the hospital you register at (or higher level facility they refer you to). If you had both SS and private insurance, SS would pay for all care at the hospital you registered at and the private insurance would cover care elsewhere should you prefer a different hospital. Note that while SS covers both inpatient and outpatient care, private policies that include outpatient cover are so expensive as to not be worth it, most people get inpatient cover only. So your private policy would cover only hospitalization at other than the hospital you are registered at under SS.

Private policies with decent level of cover do not come cheap, you'll be paying a lot for the option of being hospitalized elsewhere and still paying out of pocket for outpatient care elsewhere (free of course at the SS hospital).

Be wary of locally issued private policies with unrealistically low coverage caps, they are not worth having.

Neither SS nor private insurance will pay for VIP hospital room and the like (which do not equate to a highet level of medical care anyhow).

If it were me, I would just make sure to be registered at a good, large govt hospital under SS (the private hospitals which accept SS are in most cases small and substandard, though there are occasional exceptions). Then I'd pay out of pocket for outpatient care at times (which you'd do anyhow under private insurance) and count on the SS for hospitalization.