That becomes the main issue, the Concorde as it was had 26,400 gallons <=> 95,680 kgs of fuel, and would need more than double that (likely) to do the flight in one go (bar major improvements to efficiency, and counting that more weight at takeoff needs more weight of fuel to fly that extra weight).
A quick search says about 40 minutes to refuel a jet, so a stopover is going to add at least an hour, probably more because they have to come out of supersonic, etc.