I don't find the information printed on referee / DM screens particularly useful. I use one, but to organise information vertically as well as horizontally. Here's some notes about the way I use them:
1) I use gator clips and paperclips to attach paper and index cards to the panels of the referee screen. I attach information to both the inside and outside of the screen. The outside of the screen is extremely useful as a way of presenting information the players collectively need, though remember to print in a large enough font, or write it large enough, that it can be read from across the table.
2) I use a five-panel screen. On the inside, from left to right, I attach:
a) Random encounter tables, terrain tables or generators, and any overland travel rules I need.
b) Maps in the order I expect to use them.
c) Monster stats, clustered by encounter and then each cluster sorted by the order I expect them to occur.
d) A list of random names. I cross names off as I use them. Behind that, list of major or important NPCs, a calendar / timeline and a relationship map.
e) A list of PC stats that I can consult without telling them (i.e. perception scores, marching order). Behind that. I keep a list of treasure items and information that I expect to use in the upcoming session.
Most of these are sheets of paper. I take them down off the panels as necessary, or else I'll spread them out if I have a minute or two (i.e. grabbing all the monster stats and clipping them to separate panels temporarily so I can track the combat)
3) On the outside, from left to right, I'll attach:
a) Any procedures the party follows as a whole (e.g. overland exploration).
b) The guard's random encounter table unless their filling it out.
c) A copy of any map they have that's relevant to the situation.
d) A list of NPCs, places, etc. they've met who they should remember.
e) A list of any gear, retainers, etc. that the party as a whole has.
Players are free to take stuff off the screen at any time to either look at more closely, to update them, etc., but I try to make sure they put them back up on there before we go onto anything else, otherwise they tend to vanish. The PCs are welcome to attach any other documentation they want to the outside. I use player roles so people know which documents they're responsible for, and which are someone else's problem.
4) I use cue cards / index cards to track information I need at a glance. I tend to generate index cards during play, instead of beforehand, using a sharpie. Remember, you can clip index cards along both the top edge and the side edges of the screen.
Specifically, I write the NPC's name, what they want, and any critical stuff (e.g. "Offers quest to slay goblins - 500gp") on a cue card in black sharpie and attach it to a panel facing towards me when I'm running them. If you get caught up in an interaction or distracted and lose track of what's going on, I find this is most of the information you need to get things back on track. The other thing I commonly do is write triggers for traps, or ongoing environmental conditions on cue cards and clip them facing me so I don't forget them as we go on. If the PCs have weird powers like seeing invisible enemies, I'll also note them on an index card so I remember the invisible enemy or whatever can't sneak up on them.
On the exterior of the screen, I clip index cards with the names and 1-3 word descriptions of any important NPCs they're interacting with. If they're in a distinct location I'll add that too. I'll also clip index cards covering any ongoing environmental conditions the PCs are aware of, along with the mechanics and effects. e.g. "Dark -2 to hit". I usually use one of the side edges to handle marching order for the PCs. Lastly, I'll sometimes clip index cards with any resolutions or goals the PCs made earlier that are shaping their decisions (or any quests they picked up that they might've forgotten about). You'd be surprised at how often players forget these things, so having them written down helps.
I have the players write player-facing cards, rather than taking up my time to do it, except for ongoing environmental conditions. Usually you can dish out some markers and sharpies and handle most of the card writing you'll need in a given encounter in a minute or two if it's distributed around the table. When in doubt, it's the rules coordinator's job to update cards with environmental effects and the caller's responsible for keeping the marching order and the quests / resolutions correct.
5) I have my PCs handle initiative rather than tracking it myself, but if you haven't started doing this yet, you might want to attach it to an inner panel.
6) This sounds like a lot of work, but most of its front-loaded, and there's several ways to save time. In play, it tends to save a lot of time you'd spend wracking your brain for relevant information or reminding people of the same unchanging facts over and over again, etc. and allows you to ensure you have all the information you need right at hand. Here are some time-saving measures:
a) Use print-outs whenever possible. I print off monster stats straight from the book's pdf and write things like HP totals directly on the sheet. I grab images from the internet and print them off for maps etc. I'll print off the page the treasure item's description is on and then just use a highlighter to outline it so I know what's the right thing.
b) Appoint one of the PCs (usually the caller or note-taker) to store any cards they aren't using, and to be responsible for retrieving them when they're needed.
c) Assign a player (the rules coordinator) to produce the document with all the relevant stats you need (perception scores, stealth scores, saving throws, etc.). If they don't produce a new one, you'll go with whatever the old one says, even if it's worse.
d) If possible, seat people so they can reach out and handle the screen carefully to remove and add documentation as necessary. If not, sit someone responsible next to you and anyone who needs to change stuff has to go through them.
7) You can swap out cue cards for post-its if you find it easier. It tends to be harder to reuse post-its, so the more static something is, the better they are. If you only need something once and will never need it again, they're great. If something will come into and out of play repeatedly, index cards tend to stand up to repeated use a little better.