There could be a lot of reasons why your side character can’t give this highly important information to the protagonist right now. Maybe she has to go visit her dying aunt for a moment. Maybe she wants to go drink at the bar, thinking the info can wait. But we can guarantee that “I have something to tell you when I get back” is by and large a death sentence for that character.
Why this can be bad: Sometimes this makes sense. Sometimes. But a lot of times, as an audience member, you’re sitting there frustrated not just because this info would help the protag so much, but because it makes absolutely zero sense why the character couldn’t tell the protag right now, especially if it’s of dire importance. Why does the protagonist never ask, “What is it about?” or “Why can’t you tell me now?” And if he does ask, why is there always some bullshit excuse like, “In due time.” In due time? We’re probably talking about the fate of the world right now! Don’t give me your “in due time” crap! Tell me what I need to know! But writers like to do this for the dramatic irony and the tension, knowing readers will be frustrated. But that’s not a good enough reason to do it, not if it doesn’t make sense.
How you can fix it: If you do want to use this, you sure as hell better make sure it makes sense. Does the side character know the protag is looking for important information regarding what she’s learned? If so, why does she not tell him as soon as she learns the information? Why does she need to wait until she’s done another task? Or, in some cases, why does she insist she needs to be there in person rather than telling the information over the phone, especially if it’s a matter of life and death? Your characters’ decisions need to make sense in the context of actual human beings; you can’t just make them do stupid things because it makes for more drama. If you absolutely don’t want your character to tell this information, then there needs to be something urgent to draw them away immediately.
Bottom Line: Your characters need to make sense; stop abusing their sensibilities for drama.