Add data directory on PostgreSQL 10
As you noted, you can do this with tablespaces to split the data over multiple drives.  But using tablespaces makes things like restoration and migration more difficult.  Also, you have to actually move some objects to this space, or it won't accomplish anything.  You set it as the default, but that only applies to new objects, it won't automatically spill storage for existing objects to the new tablespace.  Any single object can not be split.

Is it really necessary?  Of the 63GB main drive, how much of that is used by the database itself versus other things?  Rather than using a tablespace, it would be cleaner just to move the entire database to the new drive, assuming it has enough space to hold it all on its own.  If most of that 61GB is for this database, then of course that wouldn't work. 
In that case, to make things easier to manage, I'd probably drop the tablespace, and ask them to create for you a single new drive large enough to hold the entire database single-handedly, then move the whole thing to it.