Hi there Both.
Sorry, I saw this yday… Ive been quietly lurking around here, but PG questions have got pretty advanced beyond my hello world …web dev skill levels.
I was going to reply yesterday and say, Ill reply properly later to you with some advice… and then the day went by! stuff happened , so I’ll say , I’ll try and reply properly later.
I was going to suggest using the dosdude1 installer and update the OS , which required knowing exactly which model of the 2013 iMac it is … Metal, non Metal , graphics capability etc.
But… when I looked, I couldn’t find DosDude1’s website.
!!!
what!?
so here is a quick search on this forum.
so this would break you free of APPLES Year of Manufacture, castrating OS Install Limitation.
and, yes I would. definitely USB install at first to check it all out, then clone the disk to a separate partition or just install it.
Ive probably got a lot of waffly install instructions on here about it.
I ll have a quick look now.
And as for recovery, Ive had success with DiskDrill -as well as such old tricks as Placing the HDD in the freezer/fridge for sometime and then retrying to recover data from it again etc.
BUT
not with Snow leopard!-10.6.8
a bug in the DD app for this OS caused it to insert it its app icon in the Mac OS file system in a location which caused the OS to become unbootable…after installing DD and trying to recover data from a failed disk!
A pretty non intuitive situation considering you were already stuffed… due to a non booting HDD!
But I digress.
And YES… for the solid state disk. Really, the performance hit is massive on older machines, but you ALSO have to make sure that the OS your running has TRIM capability enabled for 3rd party SS HDD’s or otherwise… when files are deleted er, they basically aren’t! the files stay on the SS HDD and aren’t actually cleaned away as such and you will notice a performance hit over time as the SSD clogs up with NOT Deleted files and the OS just keeps moving them about! - seriously.
there are techie explanations of this about. But basically, it was an APPLE decision to screw. you if you didn’t pay for their ULTRA expensive Apple SS HDD’s
so I’ll have a look later, if you could advise me which actual model and late mid/early part of the year of manufacture.
COOL!
just checked again!
I can connect to it now.
His installers will allow you to install up to Catalina on non supported Macs (depending on your version/graphics cards, wifi bluetooth hardware etc)
this is the way to go with regards your 2013 imac OS install
ah! ok
Corrected! Here I STAND! - you may just notice quite a bit of striking through up there 
dont bother with the Dosdude1 patcher tool as … you can run Mojave, Catalina etc.
…Dosdude1 patchers flat line just before Big Sur and never produced an installer for that or beyond.
however, paying heed to what @SilverT says, and the fact that you would have USB bottleneck - BUT STILL MUCH FASTER than standard HDD on these later OS’s
(especially if USB 3 - which is what I think your iMac would have)… the SSD would prove to be much faster an experience that spinning disks!
I now only use them for back up archiving (well and some gaming - when I used to have time)
Well… actually, Ive not even done anything remotely archivey…for aaaages.
I hope I can remember where my passwords are stored!
This following link is relevant.