Yeah , I have run into this situation. You can do it , you just need to buy the SAS cable third party as Dell does not sell the cable needed as part of the installation kit for the H710 or H310 hardware controllers and your current S110 software controller cables will not work. You  also need to know whether you have the cabled hard drive or the hot swap hard drive chassis and whether you have the 3.5 or 2.5 drive chassis. Fill me in on these 2 options and i will try to help you find parts.

Also , btw, your current software controller, the S110 will do a RAID 10 of 4 hard drives, just wont be very fast.

i.dell.com/.../dell-poweredge-s110-spec-sheet.pdf

he S110 will do a RAID 10 of 4 hard drives, just wont be very fast

There is not a situation where the S1x0 controllers would be preferable. In fact, there really isn't a situation where they should be used at all. You can absolutely upgrade to a hardware RAID controller. Dell Support probably misunderstood your question (not usual) or you misunderstood theirs (also easy to do).

You mention the S110 controller, I don't see any mention of that in the system configuration.

Here's the specs from the original system configuration page.

318-2080 : 3.5" Chassis with up to 4 Cabled Hard Drives and Embedded SATA
Part Number Quantity Description
X45GR 1 ASSEMBLY, CHASSIS, SYSTEM, CABLE, R420, V2

341-9247 : 500GB 7.2K RPM SATA 3Gbps 3.5in Cabled Hard Drive
Part Number Quantity Description
1KWKJ 2 HARD DRIVE, 500GB, EXPANDABLE SYSTEM, 7.2, 3.5, W-SU, E/C

331-6958 : SATA Cable for 3.5" in Cabled Chassis
Part Number Quantity Description
H3YKR 1 ASSEMBLY, CABLE, MOTHERBOARD, HARD DRIVE, SATA A-D, R4/320

We're looking at something like this, not sure about cabling, though. This card has SFF-8087 mini-SAS Internal Connectors, the H3YKR cable looks like it has that connector.

"Dell Support probably misunderstood your question (not usual) or you misunderstood theirs (also easy to do)."

Well, they understood you perfectly.Their absolute position as you cannot upgrade from a software to a hardware controller on a 12 or 13th Gen Server. The simple reason is that they do not sell the cable needed with the hardware controller, or separately after POS.

"You mention the S110 controller, I don't see any mention of that in the system configuration."

The S110 is a chipset on the motherboard on the R420, and judging by your order, it just was not turned on from the factory. You can turn it on by going into the Bios/ Sata Settings and set it to raid. FYI this will blow away all data if the current mode is ATA or AHCI.

Your current cable will not work, you need

Dell Part # XP00F  - DELL CABLE MINI SAS TO 4 SATA HARD DRIVE CONNECTION FOR DELL POWEREDGE R320 / R420 

The H710 you can purchase from Dell, it wont come with the cable, and you may want to get with Dell spare parts to see if they will sell you the cable. Although you may be stuck with the internet sites for the cable.

http://accessories.dell.com/sna/productdetail.aspx?c=us&l=en&s=gen&sku=342-3534

I would not recommend going with the H310 hardware controller as it has no cache and will just not perform like the H710.

Continuing...

We purchased and installed a Dell PERC H710 Integrated RAID Controller. I plugged in the original cable, an H3YKR, and all the disks came up on the controller. So we set it up as RAID 10. It's initializing at present.

Some of the cables to buy were listed as XP00F, had a picture with a tag of H3YKR on the cable.

Thanks for the update, I appreciate when folks let me know the outcome.