Coordinating moving with giving 30days notice to current landlord

Hi this might be a silly question but I am currently searching for a new apartment and am month to month at my current place. The landlord only requested that I give the standard 30 days when I decide to move on. Looking on Street Easy seems like on pointless until I actually give the notice to my landlord as I will not be ready to actually sign a new lease as they are all looking to fill the apartments by the beginning of the upcoming month.

I am worried about giving notice to my landlord and then not being able to find a suitable apartment in time for the targeted move. I guess I could always just tell them landlord that I can stay an additional month if I don’t find something suitable. But that could conflict with their own plans to fill the apartment once I move out. So kind of confused how best to approach this and am wondering if others can share their own experience. Do you just give your notice on the first of x month and hope to find something within the first half or so of the month to give you adequate time to move by the months end?

Thanks