I just successfully updated my Xcode tool to 5.1 version and when I check Downloads option in preference section then I noticed there is no option to download ios 6.0 simulator.
There are basically no difference between IOS 6.0 and IOS 6.1, in 6.1 apple just added three header file with few method in MapKit framework
and did little modification in UIKit
. So I don't think that you should bother about that.
You can check the modification here
Please follow this procedure to upgrade your xCode
Upgrade xCode
At last change deployment target to access both IOS 6 and IOS 7 simulators.