And to the topic, multiple chainrings - yes it can be done.
The Peter Mooney of my username is currently set up as a "triple chainline" fix gear with three chainrings and three cogs on the rear wheel. It is traditional old school fix gear in that it runs the stiff 1/8" chain and chainrings and cogs. Two cogs on one side of the hub, one on the other. Slight dish in the wheel so the two are pushed inboard and the one outboard. Each cog lines up with its respective chainring.
But - no tensioner! I pick chainrings and cogs where the teeth add up to roughly the same total and the hub stays in the horizontal dropout slot with proper chain slack. Chain has near perfect chainlines in each of the three combos. If I wanted to, I could do skid stops. (Don't because my knees would fail.)
OP, if you can fit a Surley "Dingle" (a two-cog package that screws onto a standard fix gear hub just like a single fox gear cog) onto your IGH, you could mount that Dingle, put on two chainrings (using my keeping the chainring and cog tooth total roughly constant) and have two different gears. (You are limited to 3/32" chain. Many are OK with that. I ran 3/32" for decades but have never regretted the switch to 1/8". My version of the "Dingle" uses a 21 tooth cog as a base and a 17 tooth cog with its center cut out brazed by a pro to the 21. Very expensive but 1/8". Works beautifully.)
And to be able to play these games with a free-er hand, have a bike with a long dropout or track end slot. (I much prefer dropouts because they make flipping/removing wheels so much easier but if you are running both cogs on one side, that isn't an issue until you flat.) The perfect tool to carry - the Pedros Trixie, a combination of 15mm hub wrench and a really good lockring spanner. That spanner allows you to lift the chain off the cog or chainring and drop it onto its neighbor with clean hands. I sew a strap and velcro onto my toolbag so I can slip the Trixie under the bag and secure it quickly. Another trick - use the fender eye on your dropout if you have one or drill and tap a track end and thread a panhead screw in from the inside to act as a chain hanger. (Very useful for wheel flips! Faster and yet again, clean hands.)
I hope this leaves you with some ideas. Fix gears are a blast and shouldn't be limited to flat ground only or young men who still have knees.