Note that the September 2023 FAQ entry stating that "If it's discovered during a skill test, it must be committed to that skill test" is not entirely accurate, at least according to this response from FFG:
Q: "Hello! If I see Surprising Find when I search my top three cards from Eureka! During ST.7, does the Surprising Find commit to the current test and draw me a card for its success?"
A: "Thank you for your interest in Arkham Horror: The Card Game. No. Eureka’s ability resolves after the point in which you can commit Surprising Find to the skill test; there is no longer an eligible skill test for it to be committed to. You’d have to commit Surprising Find to a future test. Feel free to reach out to us if any more questions arise!"
In other words, there appears to be an indefinite moment between ST.3 - ST.7 in which the current skill test ceases to be "the next" skill test.