Applying for a first time passport - might need it urgently in the coming days. Please help

My child passport expired a while back and I just forgot to renew it but now I am in a situation where I might be need to travel to Mexico in the coming days for my dads funeral. I don't know the *exact* date because there are complications with the funeral arrangements so I dont have an airline ticket for an urgent application, but Im sure its going to be sometime in July but things could change.

Do I apply for a new passport with expedited service and get my application going, and call the National Passport Information Center to switch the status to urgent as soon as I book my airline ticket?

or

Wait to apply until I get my airline ticket and apply for an urgent passport?

or some other option you guys can let me know of. Thank you