I don't think so cause cold air sinks. You would just be recirculating the cold air before it had a chance to cool down the room. Let the cool air settle to the floor and pull the warm air from the ceiling (warm air rises). Put a 2nd return in the ceiling on the opposite side of the room from the current one. If you "Y" off of the current duct going to the old ceiling register then the duct needs to be bigger than the old register (in cross sectional area) to achieve any gain in air flow. If you increase the air flow in this room then there might be an equal drop in air flow in the rest of the house. This might then mean that a larger blower motor and fan is needed.

But, what is it that you want to accomplish? Is there a cooling problem in the room?