There are many, many, many, many possible ways to route from LAX tro MSY..... I'd go so far as guarantee that a seat is available for absolutely any given date +/- 1 day.

The routing rules from expedia for LAX--MSY are:

ROUTING 784 *** TRAVEL MUST BE VIA CLT ***
784 FROM LAX-US-SFO/LAS-US-DEN-
VIA US-STL/MSP-
OR VIA US-CHI-US-IND-US-CMH-
TO US-PIT-US-MSY*
FROM LAX-US-LAS-US-DEN-
VIA US-MKC-US-CHI-US-IND-US-CMH-
OR VIA US-STL/MSP-
TO US-CLT-US-MSY*

You can have up to 5 segments, so swing through CLT and you should be good. Start by finding a T seat from CLT to MSY, and build the rest off of that.

Don't forget you can mix UA and US operated flights, as long as all are booked under a UA code for this UA fare.