Correct term for a person working two jobs secretly

The correct term for this is moonlighting, which means:

Have a second job in addition to one’s regular employment.

It often carries a connotation of secrecy. The second job is done under metaphorical moonlight. Cambridge captures this connotation:

paid work that you do in addition to your normal job, especially without telling your employer.

It fits perfectly into your example context:

"John's employer found out that he is moonlighting for other companies. Now he's in serious trouble."

Here is the etymology according to Etymonline.com:

moonlight (v.)

"hold a second job, especially at night," 1957 (implied in moonlighting), from moonlighter (1954), from the notion of working by the light of the moon; see moonlight (n.). Related: Moonlighting. Earlier the word had been used to mean "commit crimes at night" (1882).

I think the moonlighting answer is correct, but in the interest of throwing another word into the mix: you could say that the person is freelancing:

working for different companies at different times rather than being permanently employed by one company.

"a freelance journalist"

synonyms: self-employed, independent, contract

"John's main employer found out that he is freelancing for other companies during his off-time. He is now in serious trouble."