I don't have direct experience with the Shimano 6700 level components. However, a little research shows that your current RD-6700SS (short cage) is limited to 28T max large cog and 33 tooth capacity. The RD-6700GS (medium cage) is still limited to 28T max large cog, but has 39 tooth capacity. Doesn't help as you want a lower gear (larger cog). There is a 12 - 32 10 speed Ultegra cassette, which would give you marginally lower gear, but is intended to be paired with the Ultegra RD-6700A RD, which has a 30T maximum cog size.

What you probably want to do is go with a 32T maximum cog to get a "significantly" lower gear to make this worth doing, which means you would need a MTB cassette. Shimano 10 speed MTB cassettes (11-32, 11-34, or bigger) have the same spacing as the road 10 speed cassettes, so no problem there. Its doubtful you could get your current RD to work with a 32T cassette let alaon 34T. You might be able to get the RD-6700A to work for a 32T cassette (depends on you rear derailleur hanger and whether you could adjust the B screw enough that the pulley would clear the largest cog). You can always go to a Shimano MTB RD, but have to get a 9 speed model, which usesthe same cable pull as Shimano road shifters (10 speed MTB derailleurs are not compatible with 10 speed road). Fortunately they still make 9 speed RD in several upper level groups (Deore M591 or XT M771 or M772).

Good luck.