1. If a product derives from Federally-funded research, the government owns a share of the IP for that product. (This share should be larger than the monetary investment in the grants that bore fruit since the US taxpayer funds a lot of early-stage research, only a little of which will result in IP. So, this system must account for the investments that didn't pan out as part of the total investment required to produce that product.)
2. Fund grants based on models of downstream benefit. Four things that should be included as "benefits" in this model are increased health span, increased capacity for bioengineering, an increased competent researcher pool, and a diverse set of researchers. Readers from backgrounds like mine may balk at "diversity" as an explicit benefit; however, diversity is vital to properly exploring the hypothesis space without the bias imposed by limited perspectives. Edit: see the replies for a discussion of what I mean by diversity.
3. Classify aging as a disease/disorder for administrative purposes. Set the classification to be reviewed/revised in 20 years after we have a better picture. (Whether it should be considered a single disease from a reality-modeling perspective is uncertain, but being able to target it in grants will give us more research that will help us model it better.)
4. Encourage inclusionary zoning at a Federal level.
5. Create a secure government-wide password manager. (If necessary, the HHS is large enough to do this alone, but the benefit would scale if used by other agencies.) Currently, HHS passwords may not be placed in password managers, leaving the HHS open to phishing credential stealing attacks. The project could be open-sourced to allow private firms to benefit from the research and engineering.
6. Make all health spending tax-deductible, whether or not it is funneled through an insurance company. (This is probably the domain of Congress, but maybe there is something HHS can do.)
7. Reduce the bureaucracy/red tape for TANF recipients.
8. Combine FEMA and ASPR
9. Work with the Census Bureau to collect and publish statistics on human flourishing in the US and push/advertise to make those numbers top-line numbers that the electorate (and thus politicians) pay attention to. Improving these statistics can be a "benefit" in the grant funding proposal above. HHS can also work to create conditional markets to predict how different decisions will affect those statistics.

I like vaccines and suspect they (or antibiotics) account for the majority of the value provided by the medical system. I don't usually see discussion of what can be done to promote or improve vaccines, so I don't know much about it, but the important part is they remain available and get improved and promoted in whatever ways are reasonable.

Beyond that, a major health problem is obesity and here semaglutide seems like it would help a lot.

I think our collective HHS needs are less "clever policy ideas" and more "actively shoot ourselves in the foot slightly less often."