From the song: 金木犀の夜
I'm wondering if it can mean "I'll try to remember the phone number, then look (if I remembered)", cause that's what my intuition tells me right now, but I'm asking to be sure if that's eally it.
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My answer is really just an extension of this answer as it is applied to your sentence. For further details, please see that answer.
してみる mean very different things - the former means
attempt to X and the latter corresponds to the English
do X and see, as in doing something and then reflecting on the result.
Consequently, a very direct translation of
電話番号を思い出そうとしてみる looks something like
I will try to remember the phone number and see how it goes where
try to comes from
see how it goes comes from
Incidentally, there's no way to know without more context from the song, but I suspect this is historical present tense, in which case it's really talking about something that has already happened.