Probably the easiest solution would be to just chuck a larger disk in the system and retain the original drive for the operating system. If you do not need the high speed of an SSD, you may be able to get more storage space for the money by going with a spinning disk. 7200RPM drives are fast enough for most applications, though you may run into issues streaming 4K (or higher) resolution video.

Another option would be to start building out a storage pool using some type of RAID technology. On my own server, I use ZFS for the data partition. It is basically a software RAID. I use a RAID-Z1 configuration, which stripes the data over multiple disks (three in my case) and uses a parity calculation to provide data redundancy. It also has the advantage that it can be expanded to new disks dynamically and does not require that all disks are the same size. Initial setup does require more work and you are now monitoring multiple physical disks, but having a unified storage pool and redundancy is a nice way to go.

Any way you go, just make sure you have good backups. Drives fail, and sometimes even early in their life. Backblaze reports can be an interesting read when looking at drive options, as they really do put the drives through the wringer.