"Is it reasonable safe to rollover a RAID 5 into a RAID 6?" Yes, just as safe as migrating from a 3-disk RAID 5 to a 4-disk RAID 5.   RAID 6 is like RAID 5 but uses two disks' worth of space to store parity, so it will withstand two failed disks while staying alive.  The S300 cannot do RAID 6 ... it is a very limited "software" RAID controller. "Ideally, what I've always wanted to do was rebuild the array into a smaller and smaller total size so we could swap out all mechanical drives with solid state, but that looks pretty difficult to do without re-initializing the entire array." Because SSD's cannot be used in an array with HDD's (likewise SAS and SATA cannot be mixed), the only way to convert your setup to SSD would be to backup/replace/create/restore. Given your current controller, I would maintain the 3-disk RAID 5 + hot-spare.  That's just my initial thought.
We are using a PowerEdge T110 with 4 available bays.  We are currently using a RAID 5 setup with 3 250GB drives.  We have a spare 250GB drive in the 4th bay, not yet plugged in.  We would love to do a RAID 10 setup but are wary of the consequence of rolling over the entire array.  Because of the nature of the server, we don't have an easy way to rebuild it in case of a total array failure.  That said, would it be worth it to keep our setup as RAID 5 with 3 drives + a hot spare or would we be better served doing RAID 5 with all 4 drives?  This is under the assumption that we would be using Open Manage Administrator to expand the array to 4 physical disks once I plug in that 4th drive. The server is an SQL server for a PoS system.  In one article I read, the author made a good point about how a hot spare is NOT a good idea, made some sense.  But anyway, what would we be best served by?
RAID 5 using 3x250 + 1 spare or RAID 5 using 4x250?