Mario 3 hides the left 8 pixels. As rainwarrior pointed out, you can entirely hide the tile errors behind those hidden 8 pixels, even though Mario 3 actually fails to hide them.
If you don't want to hide the leftmost 8 pixels, you just offset updates by half a tile. What this gets you is the new column not being updated until it's 3 or 4 pixels on screen. The left and right of the screen would share the error area.
I can't tell if you want the left 8 pixels hidden for the final or not, but regardless you can actually just brute force pixel offsets added to the code that checks for a new column to find the best solution. There are very few values to try.
Edit: GIFs! Linked because they're biggish.
This is a 4 way scrolling engine with horizontal mirroring and the left 8 pixels hidden:
http://i.imgur.com/tvTgLtu.gifYou will never see a wrong tile on either side of the screen. Wrong palette stuff, sure.
This is the same 4 screen scrolling engine with horizontal mirroring and the left 8 pixels shown:
http://i.imgur.com/fp550Az.gifYou'll notice that there are CONSTANTLY errors on the left side of the screen. That's why it's being hidden!
This is the same 4 screen scrolling engine with horizontal mirroring and the left 8 pixels shown, with tile updates offset by -4:
http://i.imgur.com/HwHWYUj.gifYou'll see tile errors on both the right and left. You have to scroll a few pixels to get the new column in either direction. (-4 is not necessarily the right value for this or any other engine. This seems to have more errors on the left than right.) This was just a quick hack, though. A possible setback is left and right no longer want to update a similar tile. (I remember having trouble with something like that when I offset the attributes to split the error between left+right, but it has been a while.)
Anyway, I don't just mean to show, "Look what I can do!". But without seeing your code, all I can say is try to offset when the updates happen to accomplish whatever looks best for your goal (hide left 8 pixels vs show 8 pixels ). Maybe seeing the where the column updates happen in relation to the scroll position in the above gifs will give you an idea of how you need to offset for whatever you're doing.
Edit 2: The way to think about it (assuming you want to hide the pixels) is, you're at the very top left of the screen. The left 8 pixels of your level will NEVER be shown, so at that scroll position, you want columns 1 through 32 of it in the nametable, rather than 0 through 31. Then every 8 pixels, you draw a new column.
If you want to show the pixels, well, I don't have a good way to think about it because I've never done that.