Could try gently lower the card a bit then tighten the screw, enough to fit the HDMI cable, i can see there's not much room. Either that or find a HDMI cable with not so much plastic around it.
Rest of the outputs are DisplayPort and 1 DVI-D. DisplayPort is going to have the same problem. Besides altering the case with a chisel to make room for a plug which would involve taking everything out to prevent damage, only other option i can see without too much mucking around is using the DVI-D port with a DVI to HDMI cable or adapter. If your monitor has DVI-D or DVD-I, can use a DVI-D male to male cable instead. Depends if your...