Digikam is a cross-platform application that has the ability to import only images that have not already been imported. The feature is confusingly called "Download New". See Using a Digital Camera With digiKam.

To remove some of the tedium of manually organizing images, you can use exiftool to rename and organize files into folders by date-time:

exiftool -v -P -q -fileOrder FileName \
   '-FileName<${DateTimeOriginal}%-c.%le' \
   '-FileName<${DateTimeOriginal}.\${ImageCount}%-c.%le' \
   '-FileName<${DateTimeOriginal}.\${SubSecTimeOriginal}%-c.%le' \
   '-d %Y%m%d/%Y%m%d-%H%M%S' \
   -ext jpg -ext mov -ext mp4 -ext avi -ext cr2 -ext raf -ext dng .

Then you can use rsync to copy only the changed files to your hard drive:

rsync -FatHz -h --info=progress2 [source]/ [destination]/

You can write your own scripts to keep track of files that have already been copied from the SD card. For example, on Linux, the following can be used:

destination="path/to/incoming/folder/"
for file in *.jpg ; do
   if ! grep "$file" list.txt ; then
      echo "$file" >> list.txt
      cp "$file" "$destination"
   fi
done

This approach can be improved by tracking checksums or other file metadata to avoid problems with filenames repeating at 0001.