It is completely possible that the update added a lot to the device. Even if you don't 'see' it from your end. Plus anything you use on the device will add to the data. Say "Facebook" or a "Browser" all add data to the phone, so unless that gets cleared out, it takes up even more space. Which isn't stuff that is normally moved or removed. You have to go into the phones 'settings', 'apps', 'all' and then locate the individual app to clear its cache or even all its data. Just be sure you are OK with removing whatever it has before you do it.
The only other thing you can try and do is to see if the phone will allow converting the SD card to "Internal" rather than "External" storage. Not all phones allow for this though, and you have to start with a blank SD card, if it will allow it, as it will overwrite anything already on the card.
First, you need to be using a phone that is running at least Android "Marshmallow" (or newer) or be able to upgrade to it. Second, you will need the instructions I am including below. And third, you need a phone that has not been restricted by the manufacturer from making this change. Which does happen.
NOTE: If any of the steps are missing on your phone, or they don't work for you, then you have a phone that either has an older Android version or has been restricted by the manufacturer.
How to turn the "External" SD card into "Internal" storage on "Marshmallow" a.k.a. 6.0.
NOTE: Please be sure you start with a blank SD card. It needs to be blank because when the card is converted to "Internal" storage, it will be formatted (wiped) and encoded (making the cards data only readable by that device).
1. Go to device “Settings”, then select “Storage”.
2. Select your "SD Card", then tap the “three-dot menu“ (top-right), now select “Settings” from in there.
3. Now select “Format as internal”, and then “Erase & Format”.
4. Your SD Card will now be formatted as internal storage.
5. Reboot your phone.
NOTE: If you don't reboot the phone, many things may not work correctly, so make sure you do.
How to turn the "External" SD card into "Internal" storage on "Nougat" a.k.a. 7.0.
IMPORTANT NOTE: Be sure you start with a blank SD card. When the card is converted to "Internal" storage, it will be formatted (wiped) and encoded (making the cards data only readable by that device), so anything on it prior would be gone.
1. Go to "Settings", and then select "Storage & USB".
2. At the bottom of the list you should see the SD card's details, including the option to format it and make it "Internal" storage.
3. Once this is done, reboot the device and you can start running things from the card.
If you don't reboot the phone, some things may not work correctly, so make sure you do the reboot.