I can give you a bit of a more frank, less technical and comprehensive answer that may help. If this helps add to the other answers, great. If not, ignore me.

Adjuvants work locally as well as systemically, but not both in equal measure. They are especially active at the site of the injection. A vital part of the desired vaccine-induced immune response relies on the adjuvant and vaccine antigens being in relatively high concentrations in the same place in the body.

If you just injected a person with an adjuvant only and they already had the virus on-board, you may see a very mild increase in overall immune response in the form of a slight elevation in body temp and very, very mild systemic inflammation, but it wouldn't be nearly enough to affect the course of the infection. You would also have a local inflammatory response at the site of the injection but without the vaccine along with it, you just get a sore arm with no benefit.

The adjuvant triggers local inflammation which calls white blood cells to come running, or rolling and oozing as the case may be. These white blood cells (macrophages/dendritic cells/Antigen presenting cells) are stimulated by the adjuvant to grab the vaccine antigen particles that are highly concentrated in that area and (long story short) present those viral particles to the immune system so they can pump out new lymphocytes (another kind of white blood cell) to target and kill specifically the strain of virus that was in the vaccine.

I don't know if that clarifies or obfuscates things!

Edit: typos, clarification