Welcome. Allow me to (gently) nudge you: Consider removing the qualifier to look for other gospels in the NT itself. A good answer might make the case that the opposing gospel came from outside the Church. For example, Caesar's campaigns were proclamations of the Gospel of Salvation and Peace. I suspect that in some places, Paul was challenging which Kyrios (Lord) someone in the Roman Empire should primarily have allegiance towards.

To be clear, though, reframing the question is entirely optional. Plenty can be answered from within the NT itself (e.g., the Judaizing gospel; see Galatians 1:8-9).