Can I fly under a solar eclipse to lengthen its effect?
The speed at which the shadow of the Moon crosses the Earth varies according to time of day and latitude. However, 1500mph is a good starting point.

The only supersonic commercial aircraft to enter service was Concorde, top speed around 1350mph. And, of course, Concorde is now out of service.

Flying something subsonic along the path of totality will give a longer view, but most airliners top out at around 600mph.

You've really no chance of keeping up with it. You're probably better off finding a spot with a good view and enjoying your 2.5 minutes!

Nasa have a range of materials dedicated to this. If you fancy a little mental gymnastics  take a look at https://eclipse2017.nasa.gov/challenge-5-%E2%80%93-estimating-speed-lunar-shadow. In particular, download the PDF worksheet and do the maths!