A good technique for removing 50-pin SMD CF/MD card slot with Hot air
I would remove the part in a way to concentrate on saving the board with minimal damage as opposed to trying to remove the whole socket in one swoop. If you reference the ELCO / AVX data sheet for the 5620 you will note that there are three main attachment regions for this part - The pin row and then one anchor foot on each side toward the front of the socket. First snap off the socket cover and ejector mechanism and set aside (you may be able to re-use it with the replacement socket). Then cut the side slides of the socket at the points marked in the picture below. Cut in a way that you do not put any stress on any of the solder pads. Once the sides are cut it is super easy to remove the two anchor foot pieces using a regular soldering iron.



Next I would cut off all the solder legs across the back of the socket. My tool of choice for that is to to use an abrasive cutoff wheel in a rotary tool like a Dremel. Cut along where I show the green line in the above picture. Use care to isolate the area of the the board so that the particulate from the cutting area does not get all over other parts of the rest of the board. This can be achieved by creating a mask around the socket using plastic electrical tape and attached pieces of paper.

Cutting like this eliminates almost all stress on the solder legs to the pads on the board. Once the legs are all cut the rest of the socket is easily lifted off the board. Clean up the dust from cutting and then finally remove the mask and electrical tape. At this point it is a piece of cake to remove each socket lead from its pad using a soldering iron and a pair of tweezers. Finally come back with solder wick and soldering iron to clean up all the pads and prepare for installation of the new socket. 

Note that replacement sockets are available from Mouser Electronics for very reasonable prices.