Short Answer: Yes
As long as the ship moving into the Drift is stationary and with it's engines off, it can enter the drift inside a larger, stationary body like a starship or base. With one catch...
I can appreciate why this is a tough question. Paizo mentions the Drift drive in several sections of the core rulebook. But in most of those instances, there is a caveat sentence after the many references across the rulebook:
For information on the Drift, see page 290
There are two places where the one of barely a handful of written rules for using the Drift drive is found on pg 298 under Drift Engines for Starship Creation, and on pg 291 for the rules for Space Travel for Drift Navigation. And that rule is:
For a starship to engage its Drift engines to either enter or exit the
drift, it must remain stationary with its conventional thrusters
turned off for 1 minute.
So as for a larger "host" ship carrying the subject ship trying to enter Drift space, that ship must also be stationary. However, having the host's engines on or off should not affect the jumping ship, only if it would make them not stationary anymore.
This can also seem to work in the reverse. There is another rule for Drift use on pg 291:
Additionally, since the Drift is a plane that you're traveling
through, it is possible to pause midjump, and even to land on one of
the floating chunks of terrain or engage in starship combat
So this means that it is also possible for a host ship to halt to a stationary position in the Drift, and a ship inside can spend a minute to charge their drive to leave the Drift altogether, including leaving the inside of the host ship.
Here's the catch
There is one very obscure feature of the Drift drive, which is buried in the last paragraph on the introduction part of pg 290:
... use of the Drift does come with a catch. Every time a Drift engine
is used, a tiny portion of a random plane is torn from its home and
added to the Drift, set to float there for eternity. The farther the
jump, the larger the chunk of material, which sometimes appears near
the jumping ship, adding an element of risk: you never know when a
long jump might tear away a chunk of Hell and leave you flying through
a cloud of furious Devils.
This also doesn't offer some clear rules, but it does try to warn of some risks. If a host ship is in the Drift, and a ship inside it uses their drive to exit, there is a chance that a piece of a random plane is deposited where that jumped ship was. Also, there is a chance that when the host is in normal space, and the ship inside jumps, that piece of random plane may be the normal space. Essentially tearing a hole in space inside that space inside the host ship.
As for what that could mean... the only example is a ship making a long jump upsetting some involuntarily stowaway devils. It is up to the DM at that point. If you would like some inspiration, both the shows Farscape and Battlestar Galactica showed a smaller ship making a jump inside the hanger of a larger vessel.