After you clone your HDD to your SSD, I recommend that you disconnect that HDD and just leave it in your desktop case for a while. It is a complete backup of your system up to the point of cloning. So, if you think for any reason that something was missed in the cloning, that's a place you could go look for it.
After a while (your judgment how long to wait) you can decide you are completely satisfied that the SSD is doing everything you need and you don't need ANY of the old data on the HDD. At that point you should reconnect the HDD (this time it will probably be on a different SATA port, but that does not matter). The best options are both aimed at completely wiping out the old data and yielding a new empty HDD to use.
1. Best option, I think: download a utility able to Zero-Fill your old HDD. If your HDD is by Seagate, get their Seatools; if by WD, get their Data Guardian. Or I believe DBAN is a free third-part shareware tool that can do this. This process writes all zeros to EVERY sector of your HDD, completely wiping out old data. MAKE SURE you do this on the old HDD, and NOT on your new SSD!! But it has an important side benefit. In modern HDD's, the HDD unit itself has a background routine it runs every time a write is done. It then re-reads the sector and analyzes the signal strength it gets. If there is any weakness, it treats that sector as bad. (Now, in "normal" use, what it would do then is a series of re-reads to try hard to recover that data completely from that sector, but for a zero-fill it may not bother.) When it detects a "bad" sector it marks that in its own tables so the sector is never used again, then it allocates in its place a known-good sector from a stock of spares it has from when it was first manufactured. It keeps track of these actions and, if too many such substitutions occur, it sends out a warning message through the SMART system because its stock of spare good sectors is getting used up. That's when you buy a replacement unit and clone the data from your old drive while it is still working OK. NOTE that this process is entirely within the HDD. Windows never knows about this, and it has NOTHING to do with the CHKDSK way of checking a hard drive for "bad sectors".
Anyway, by doing Zero Fill on your old HDD, you trigger this self-diagnosis and fixing process for EVERY sector of the unit, since every sector has zeroes written. It takes hours to do this on a big drive! When it's done (assuming no disasters), the HDD has ONLY good sectors available to the outside world. To Windows, for example, it should have no "bad sectors" to find. It is also completely empty just like a brand new HDD. So when it's done, you will have to use Disk Management to Initialize that unit (Create a Partition and Format it). Remember that it is for data only, so you do NOT need to make the Partition Bootable. When that's done, back out of Disk Management and reboot, and your "new" old HDD will show up in My Computer ready to use.
2. You can skip the Zero-Fill process if you want. You can just go directly to Disk Management and RIGHT-click on the existing Partition(s) and Delete it (them) until there are none. Then do the Initialization, etc. This will still yield an empty "new" drive ready to use, but you MAY not have found and replaced all the "bad sectors". However, during the Full Format process (not Quick Format), Windows does its own testing like CHKDSK and keeps its own records of any "bad sectors" it finds and marks them in its own files "for never use".