Most long distance Blackbird flights involved taking off at low fuel weights, then refueling shortly after take off. So yes, the record probably involved refueling over the Atlantic.
Then, if they got a full tank of JP-7 around NYC, that gives them a one-way range of 3250 miles, which is less than the 2999 nautical miles of LGA -> LHR. The SR-71's range is a hair over 3200 nautical miles. Getting a full tank of the special JP-7 or landing around LHR would be do or die time, because that would be just about bingo fuel.
