yes another cable will do.
if the freesat hd box is upstairs, you'll need to run a long hdmi cable to your screen downstairs to view HD
my house has a Lexcom "home entertainment network" system using CAT7 cables and sockets. it can be used for sharing the Tv aerial, phones, broadband, pc-networking etc. and it can share AV equipment.
but, i dont know if it can share HDMI, i'll have to look into it.