What is the most efficient way to promote my resume as a graphic
artist and web designer to make it visible to a very high number of
related companies/recruiters?

I recommend looking at "trade" publications and organizations who have
the best headhunters write for them and advertise in them. A couple of
suggestions: Communcation Arts magazine where you will also find the
Creative Hotlist, AIGA, SIAA Jobs Network. Look at the companies,
portfolios and their locations to pick the places you think are ideal
for you, then write and apply directly to those places you want to
work. For companies that do not advertise, register with good
headhunters and post your work or at least a link to your web site at
services like the Creative Hotlist. Join an organization like SIGGRAPH
and participate in their yearly conference -- network!