It really depends on how nice your old Hyperspin install was. If it has complete artwork for all the games in it as well as video previews, then I would buy an
https://www.amazon.com/SISUN-Docking-Station-Reader-Black/dp/B009F7TXMK/ (powered IDE dock) and retrieve all of it. If it sucks or supports only 13 games then don't bother.
Down to the games, again, if it has 13, who cares about the old ones. If it was decked out with a full set from the time and worked perfectly then you should go with the IDE dock and retrieve them. The ROMs are a legal grey area and you aren't going to be able to easily recreate what you had from scratch. There is also a matter of size, a current full MAME romset would most likely be in the 135gb range and again not really legal unless you own all of the cabinets. A fully supported Hyperspin install with no emulators or games can alone be 15gb made of tens of thousands of support and media files in very specific places, each of them put there by hand.
Long story short, it seems like you don't really have a lot of experience tracking all that stuff down and merging it altogether, so knowing what I know if I were you I would definitely copy the elements of your old install onto your new computer's hard drive.
You should know that you will not be able to pickup everything and just drop onto your new hardrive. You will have to figure out where your MAME files and folders exist as well as where your Hyperspin files and folders exist then copy just those to exactly where they existed in the old install. In other words, if those were in a folder on C: named "dougs stuff" then they need to be put in the exact named folder in the same place on your new one. If they existed on a second partition named E:, then you will need to resize and shrink your existing new drive to make room for the secondary drive and then create a secondary partition with the same letter E: and format it. Then you can copy the old files to the exact same structure on it. If you try coping everything and drop it onto the new one, you will break the Windows install of your new computer.
Now if your old one booted quickly and straight to Hyperspin with no Windows branding logos and no action on your part, then you are going to need to do some work to get it back to that state. It will be easy to get Hyperspin to load automatically, it will be a whole other thing to keep the illusion of nothing but an arcade cabinet.
There is an option that you might have overlooked, using an adapter to install the old drive into the new box, if the old OS was Vista or newer it may just boot and then require you to provide a bunch of new drivers. If it's older (XP or older) it will require some work from a pretty savvy computer technician to boot to anything other than the Blue Screen of Death. Basically XP installs one singular "Hardware Abstraction Layer" file set based on what type of CPU hardware was there when it was installed, and it won't work with anything different. So if you go this route, ask the technician to manually extract and replace the old HAL files with the versions compatible with your new CPU architecture. It is possible, just complicated.
Good luck.