I don't believe it is possible to set header information this way. It also arguably shouldn't be possible in the first place.
If this information is really so important to you, a better way to do this might be (mis)using the recipient address. Say you use an address like
contact.website@example.com
this method is
100% compatible with all E-Mail clients
hard for the user to change (as they can't be sure whether their message will arrive if they do)
infinitely extensible - you could go as far as storing information about the user's visit in a database, and send a random ID in the E-Mail through which you can connect the web site visit to the E-Mail:
contact.website.1ahGfd23@example.com