A few years ago a TV presenter got caught out by this, she posted a cropped photo of herself online, but the original, uncropped version could be recovered from the file which showed her topless.
You can use 'exiftool' on Linux to manipulate the EXIF data, get rid of the thumbnail, or set it to whatever you want. Imagemagick's 'convert' command also has a '-strip' option that will get rid of all EXIF data including any thumbnails.