As other answers have rightly pointed out, "Air Force One" is not a particular airplane, but rather just the callsign for any U.S. Air Force aircraft on which the sitting President of the United States is currently onboard.
The aircraft that are practically synonymous with the name "Air Force One" are the iconic Boeing 747 aircraft operated as VIP transports by the U.S. Air Force and designated as the VC-25. There are two VC-25As in service, so, if the outgoing President takes one, the incoming President would still have one available, assuming neither was currently down for maintenance. If neither of these were available, there is also a fleet of four Boeing 757 VIP transports also based at Andrews AFB, which are designated the C-32. While it's less iconic, it's actually pretty common for these to serve as Air Force One, particularly when the President needs to fly into a field with a runway that can't accommodate the 747. The 757 has amazing short-field performance for an aircraft of its size, so it can fly almost anywhere that doesn't require a helicopter. And, of course, when a helicopter is needed, there's also a fleet of those operated by the Marines for VIP transport, which the President uses commonly. So, the incoming President would not be short on aircraft choices even if the VC-25s weren't available.
That being said, it's actually not petty at all, but commmon, traditional practice for the outgoing President to use one of the VC-25s to leave Washington, DC when his successor is sworn in. Former President Obama used it to fly to Palm Springs, California when President Trump was sworn in, though the flight ended up diverting to March Air Force Base due to bad weather in Palm Springs on arrival. Former President George W. Bush used it to fly to his childhood home of Midland, Texas after President Obama was sworn in. Of course, the callsign of these flights is not Air Force One, since the sitting President isn't onboard.
As a side note, if the sitting President is flying on an aircraft from a different branch, the callsign will be "[branch name] One." For example, "Marine One," "Navy One," "Army One," etc. If the sitting President is flying on a private aircraft (for example, if Donald Trump had flown on his own 757 after being sworn in as President,) it would be "Executive One."