Best I've done with a straight (non ER) 55 was 4 hours 20 minutes, landing with 1400#.
First I piss the line guys off by asking them to fill the wings first, then do the fuselage over the top. I've been able to cram the published 6690 in there that way as opposed to single point 6540. Every pound you can stuff will make the difference if you get descended early or vectored halfway to Hawaii going into LAX. Then I give the line guys some beer for their trouble.
Anyway, here's what I used to do. Climb at 250kts/.70 to 390 or FL400 and accelerate to .76. If they hold you down and level you off low throughout the climb, pull the power back to about 750-800pph assuming they don't need you fast.
Once you get to cruise alt. check the temp and weight to see if/when you can get to FL430. You are going to trade TAS for altitude, especially if you go up early, so no rush right away. I know the long range figures are M.73, but if you are fighting 80kts on the nose trying to go west and you are at .73, you are really not doing yourself a favor.
If you can make 430 and accelerate, do it. you should be more than half way through the trip at this point and once you are up there you ought to be doing around 550-600pph. If the winds die down and you are making good groundspeed, you could try .73 for a while and see if that helps. If you have a rockin' tailwind, yeah pull it to .73 and let the wind do the work.
I noticed that by climbing to 430 at the right time, you will save about 200-300 pounds of gas at .76M.
One the descent, don't come down until they yell at you, and if they bring you out of altitude 250 miles from Vegas or some crap just keep .76 on the descent and whatever power setting gives you the minimum rate like 1000fpm unless ATC wants something specific.
If they level you off low and far out, don't go 350kts thinking you'll get there faster. You'll be sucking down 1400pph at that clip and you've just eaten all of your fuel savings. Just do whatever gives you 850-900 pph. Should be about 275kts if I remember right.
Now, this is if you are trying to save every drop kind of stuff. I look at the FMS about 1.5 hours out, and barring a radical change in wind, if I just don't like the numbers and I know we are going to get screwed on an early descent or lots of vectors, I'll just stop for gas and tell the pax that before we leave home.
I never had to stop though. I did STL-SEA, STL-SFO, STL-CRQ/LAX. All can be right around 4.2 hours depending on the wind.
With the extra 359# of gas in the LR55-ERs you can get about another 25 minutes or so and land with reserve.
Interestingly, the LR-45 carries 500 pounds less gas than the 55, has the same MTOW and engine output, but it can go straight to FL450 and then step to 470. (The wing doesn't quit at 430 like the 55.) The fuel savings offsets the loss of TAS and we end up being able to match the range of the straight 55, and land with similar reserves.