Disclaimer - these are from the square range:
Normally a top off or load from empty is done just as you mention, right handed into the tube, port arms or low ready.
In a combat situation, you can also load an empty rifle over the top while shouldered and on target. With the lever down and the bolt open, you can drop one in the top on a Winchester or over to the right side on a Marlin with the left hand, close the bolt and be live while you load the remainder with the right hand into the tube. It takes some practice, but my then teenage daughter managed it quite well.
I've seen some lever gun shooters shoot left handed. This may have been to facilitate the loading into the tube with the weak side right hand. It may also have been to transition to the single action revolver on the strong side.
Like any platform, practice practice practice.