You might do an AA flight from London to the West Coast, then over to DFW. That would add up to 2,000 miles and keep within the 6-sector US restriction.

Also, if you return to LHR by going HKG-SIN-LHR, you'll add about 2,300 miles. You could also add a quick HKG-NRT-HKG if you have 2 spare days for another 3,684.

Excellent point above on the LHR taxes - it gets VERY expensive.

I seem to remember a restriction about not returning to your country of origin, so back through ZRH from DXB might be a problem

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- Bob