The simplest way to recover the HTML from an existing signature is send yourself an email.
You can compose a new email, addressed to yourself, as follows:
After receiving the email, you can view the raw email source and extract the HTML that way (removing the $$$$ delimiters). Oftentimes, the relevant part of the email source will be encoded in a supported format like quoted-printable. This can be decoded back to standard HTML using a number of the free converters online.