…that he felt like…?!
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I don’t know what the sentence in your target language is, but “feels like” could still be correct, if the going out is yet to happen. Imagine we’re a group of friends, I spoke to Tom first, now I’m speaking to you about going out later, and I will tell you “Tom said that he feels like going out”.
If using “he felt like”, the moment of going out has already passed, or Tom has changed his mind since feeling like that.