First, on the old hard drive, let's find your Firefox profile folder:

https://support.mozilla.org/en-US/kb/how-run-firefox-when-profile-missing-inaccessible

http://kb.mozillazine.org/Profile_folder_-_Firefox#Navigating_to_the_profile_folder

https://support.mozilla.org/en-US/kb/profile-manager-create-and-remove-firefox-profiles

https://support.mozilla.org/en-US/kb/back-and-restore-information-firefox-profiles

After, copy the files onto the new hard drive:

When you are ready, we can start moving parts of your old data and Copy it to the new profile.

First; open your file browser to the New profile;

Type about:support<enter> in the address bar.

Under the page logo on the left side, you will see Application Basics. Under this find Profile Folder. To its right press the button Show Folder. This will open your file browser to the current Firefox profile. Now Close Firefox.

Windows: Show Folder; Linux: Open Directory; Mac: Show in Finder

Linux: Under the page logo on the left side, you will see Application Basics. Under this find Profile Directory. To its right press the button Open Directory.

Second, open another file browser to the Old Firefox folder on the old hard drive

Warning ! ! Only Copy the files you are told. We will avoid anything that might cause problems. Here is the list of files;

cookies: cookies.sqlite Do not bother. Cookies will be created when you return to the website. However, you will have to log in everywhere. And you may lose any settings that websites store on your computer.

extensions(add-ons, themes): it would be better to download them fresh. If you need to remember what add-ons you had, open the extensions sub-folder and look at the .xpi files

logins.json and key3.db [v58+]key4.db : These are the password files. Copy Both !

persdict.dat : custom dictionary - words you added to the dictionary

mimeTypes.rdf [v55+]handlers.json : What to do with what type of file.

places.sqlite : History and Bookmarks.

prefs.js : Your settings. Note: If you were having problems, this could be contaminated.

cert8.db [v58+]cert9.db and possibly cert_override.txt : Stores all your security certificate settings and any SSL certificates you have imported into Firefox. If removed, the cert8.db, cert9.db file will be replaced.

search.json.mozlz4 : file stores the user-installed search engines that are available in the Firefox v47+

search.sqlite file and searchplugins folder store the search engines that are available in the Firefox Search bar before Firefox v47

Copy these files from the old to the new. You may get a message that the files already exist. Choose to over-write for each file.

You usually need to set Windows to show hidden files and folders in order to explore up the "AppData" path to Firefox profile folders:

https://support.microsoft.com/en-us/windows/show-hidden-files-0320fe58-0117-fd59-6851-9b7f9840fdb2

You'll find a folder named bookmarkbackups that contains several compressed data files. These cannot be read directly, except by Firefox's Restore feature: Restore bookmarks from backup or move them to another computer.

If you want to convert one of the backup files to an HTML Export-style file (sometimes called bookmarks.html) so you can import it into a different browser, I have a tool for that, but probably you won't need it: https://www.jeffersonscher.com/ffu/bookbackreader.html