it normally helps to install win 10 with only 1 drive in PC as it likes to share itself around
put the samsung in and the hdd
go into disk management and show me a screen shot of all columns in top and bottom areas (need to upload to an image site and show link here)
my guess is the hdd partition is marked as system drive - this means boot partition
even though there is just 1 partition. Windows, if its MBR, doesn't need a separate boot partition, you can make it all one partition. I had it like that on my last PC. Only restriction with mbr is the boot partition needs to be 1st on disk, and if its only got 1 partition, it does that.
note 2nd partition on C above later created by a windows update, originally it was all 1 partition.
if it is marked system drive, there are ways to create a boot partition on new SSD but if its brand new, easier option is just unplug power from all other drives and reinstall win 10. It might even use GPT instead of MBR.
Otherwise, i wonder where your search index is - this shows how to change but it can be used to just find it -
https://www.tenforums.com/tutorials/59016-change-search-index-location-windows-10-a.html#:~:text=The search index data files,locations as the index location.
as I would have thought that be on the ssd, not on hdd.
have you run chkdsk on the hdd recently? what make/model is the drive? Might help to check its health.