There is no simple answer, but in general, this is very unlikely. In theory such reinfection is conceivable, but there are no documented cases of it.
The possibility would depend on the organism and how the person’s immune system responded to it. If a pathogen is on the skin a long time, the person has probably developed immunity to it. The same holds true for an infection from which a person is recovering, so it is difficult to “reinfect” yourself unless it is a totally different organism.
Even if a person has two cases of flu in the same winter, the second case would probably involve an unrelated flu virus that was present at the same time.
The same patterns of disease would appear, so it would be impossible for the patient to distinguish between the two infections.