Your current plan is the route I'd take too, though if I may get on a soapbox for a moment:
This very thing is why every reasonably modern server I've seen from a quality OEM has 6 or more drive bays - 2 for a mirror RAID drive to put the OS and installed programs on, and 4 for a RAID 5/6/10 with your data. So you don't have to reinstall everything when you need to add storage capacity.
It's not 1999 anymore, don't use RAID 5 in production. Upgrade it to a RAID 6 or 10. (While you're at it anyway, right?)
Anyhow, another option is to force the RAID array to rebuild onto bigger disks by pulling the disks one at a time, replacing them with a higher capacity disk and waiting for the rebuild to complete... until all your disks are the higher capacity ones. At that point, with most RAID cards, newer, good RAID cards, anyway, you can expand the array to include the rest of the useable space on the drives.
It can be problematic because it's RAID5, so there's a chance of UREs (Unrecoverable Read Errors), because rebuilding from parity takes a long time, so you'll probably be waiting ~one work day per disk for the rebuild, and because the RAID card in question may or may not support this, or may not support this well.
In my experience, it's usually easier, faster, and much less hassle just to do what you've proposed.
The other advantage of replacing the whole array at once is that if things go bad, you can always put the old disks back in and return things to the old state very quickly. If you rebuild the array one disk at a time, that's usually not an option, because you'll end up with inconsistent data on the original disks and won't have a working array if you pop them all back in. Don't underestimate the value of "playing it safe" in this profession.