But does it have enough inertial speed to achieve orbit once the air runs out for the engines?
Add a couple of oxygen tanks that kick in when the O2 is too low for combustion.
A few years back they had designs for aircraft with jet engines for lower altitudes and then hydrogen fueled pulse jets for higher altitude which left the "doughnuts-on-a-rope" contrail.
Combine that with the Burt Rutan / Scaled Composites 'SpaceshipOne' shuttlecock design and you have an aircraft capable of departing and returning to the atmosphere.
https://www.space.com/16769-spaceshipone-first-private-spacecraft.html