Had to break out my logbooks for this one - I've done quite a lot of flights in the CC propliners.
My flight with the longest leg would have been Air France 271 (1958 timetable), which is a Lockheed Starliner flight. The first leg was Tokyo/Haneda to Anchorage, which was a 10 hour, 14 minute flight (3011 nm), marked by an engine failure on approach into Anchorage. (This was the result of using an older version, Volker later helped me finding the cause and remedy to the problem). The second leg was Anchorage to Paris/Orly, a 15 hour, 4 minute flight (4089 nm). My notes show me I used 31629 lbs on the first leg, and 44330 lbs on the second - both flown in real-time.
My longest continuous flight with stops would have been KLM 845 (1957 timetable), which is a Lockheed Super Constellation flight. That was a eight-leg flight from Amsterdam/Schiphol to Sydney/Kingsford Smith, via Frankfurt, Rome/Ciampino, Beirut, Karachi, Bangkok, Manila and Biak, again in real-time. My notes on that flight show following data:
EHAM - EDDF: 189 nm, 1 hour 39 minutes (due to Go Around), 5393 lbs used
EDDF - LIRA: 522 nm, 2 hours 43 minutes (again Go Around), 9131 lbs used
LIRA - OLBA: 1184 nm, 4 hours 4 minutes, 13929 lbs used
OLBA - OPKC: 1736 nm, 7 hours 13 minutes, 23464 lbs used
OPKC - VTBD: 2001 nm, 7 hours 47 minutes, 27511 lbs used
VTBD - RPLL: 1190 nm, 5 hours 18 minutes, 18127 lbs used
RPLL - WABB: 1298 nm, 5 hours 32 minutes, 16393 lbs used
WABB - YSSY: 2134 nm, 9 hours 56 minute (due to Go Around), 28131 lbs used
Around the beginning of last year, I did a number of Atlantic crossings as well - Copenhagen - Hamburg - London - New York per Pan Am Boeing 377 Stratocruiser; Tel Aviv - Rome - Paris - London - (Reykjavik) - Gander - New York per El Al Lockheed L-149 Constellation; New York - Boston - (Shannon) - Paris - Rome - Tel Aviv per Air France L-1049C Constellation; Tel Aviv - Athens - Rome - Zurich - Paris - Shannon - New York per TWA Lockheed L-749 Constellation; as well as a San Francisco - Honolulu - Canton Island - Nandi - Sydney flight per Pan Am Boeing Stratocruiser, but unfortunately I can't find my notes on those flights at present...