I agree it will be far easier to just keep the old passport until those flights are over and done with. I changed my name by deed poll when I got married and didn't change my passport for about nine months afterwards because I had flights already booked.
I notice, however, that the OP says "She is required to apply for a new passport in her new legal name". Is there some obligation here that hasn't been made clear or is she just mistakenly thinking that because she's changed her name she must change her passport immediately?