You do not mention in your post where you are travelling from.
To and from LA from ATL/DTW/MSP hubs are tough, MEM, SLC, CVG much easier. JFK upgrades to Business Elite only clear at the gate for complimentary upgrades.
Best chances for upgrades from hubs to LA are on Saturday, Tuesday, and Wednesday flights. I have seen Sunday upgrade lists from LA to ATL with over 60 passengers on it.
Depending on where in LA you need to travel and if you are going to have to connect and upgrades are important, BUR/ONT do sometimes work better.