I have not yet had to expand my Synology storage, but here is my thinking:
If you have enough in the Synology that you can afford to delete, eg you really dint need it, then clear up space in that manner,
If you want to keep everything on the Synology, then you could either purchase an expansion chassis or upgrade your drives. I would lean towards drive upgrades. At this point, the prices for 6 and 8 TB drives have come down quite a bit.
You can replace one drive at a time, let the raid rebuild, then replace the next. How this works out depends on your RAID configuration. For instance, if you have a RAID 5 setup, you will need to replace each drive and then once all are replaced, expand the volume set. If you are using SHR, the Synology should be able to expand as you replace each drive, BUT it will not be expandable until at least two drives are replaced because of how SHR works.
I believe the help system on the Synology provides a good step by step walkthrough for the process.
If you add an expansion chassis, I believe you could either configure the drives in that as a separate volume and create new shares or it, or add those drives to the existing volume on the current drives.
My worry about adding the drives to the existing volume is that the expansion chassis connects via an SATA external cable. If for any reason that cable failed or was disconnected, that would mean that multiple drives would be lost for the volume. I have no idea how or if the Synology can handle that occurrence. I assume it would be the same as having SHR or RAID 5 with one drive redundancy and having two drives fail simultaneously, which would cause the RAID array to fail and likely not be recoverable. I don’t know this for sure. I have not seen anything online that answers this question.
As a result, if it were going with a pen external drive chassis I would make the drives there be a separate volume so that if the cable got bumped, the volume would go offline but likely be recoverable with limited data loss.