So it sounds like when you reformatted your microSD card that 'broke' the sync function between your phone's Google Photos app and your online Google account (where your photo library is stored). Going by what you described at least your photo library is more or less intact, with your previous photos still safe online but not on your phone, and more recent photos residing both on your phone and online.
So the first thing you must do is go to
https://photos.google.com and confirm
all your photos are there. If you already did this previously, double-check, don't assume anything, and make sure everything is copied into your online Google Photos collection even if you have to manually copy photos into it. You need to have all your photos in a single location at some point. Once done, you should make a manual backup of your photo library, go to Google Takeout (this will be easier on a computer browser) and make a backup of your photos:
https://takeout.google.com/settings/takeout
Save the resulting file in a safe place.
-- On your S3, with your photo library residing in your online Google account and in the manual backup file you just made, for a clean start go ahead and delete those photos that are currently residing in your phone's internal storage. At this point all your photos are residing online in your Google account, none on your phone.
-- Go to your Settings >> More >> Application manager and find the Google Photos app. Click on the 'Force stop' button and click on the 'Clear data' button. This will wipe its cache and all the Settings you made, essentially this returns the Google Photos app back to when you originally installed it. Don't start the Google Photos apps up yet though.
-- You now want to start up your Camera app, go the big gear icon in its upper left and scroll down to the 'Storage' option to choose whether you want the default storage area for your photos to be 'Device' (your S3's internal storage) or 'Memory card' (your microSD card).
-- Now go ahead and start up the Google Photos app. Since you previously cleared its settings and configurations it will be starting up the same way as when it was first installed. Be sure to go into its Settings and change things to how you want them, especially the 'Backup and sync' options. At this point both the Camera app and Google Photos app are using the same default DCIM folder again, the one that you selected in the Camera app's Settings.
-- If Google Photos doesn't start downloading your photo library into your S3 automatically (which can take quite a while depending on how many photos you've accumulated and the bandwidth speed of your online connection), you'll need to restore your photo library manually using the backup you made using Google Takeout (open the archive file and copy all the photos into your phone's DCIM folder).