I haven't heard much from Velognome in a few years, but I am running that same 2x3 setup on my Fothergill. It's pretty straightforward.

If using an AW hub, the best setup has a big cog with three more teeth than the small one (so, 18t and 21t, for example). That gives you a pretty even spread of six gears. With an FW hub the spacing is a little closer, and a two tooth difference may work better (so, 19t and 21t, for example). You can use 1/8" chain, or 3/32 if your cogs and chainwheel can handle it.

If you look closely at the teeth of multiple cog clusters, either freewheel or cassette, you'll see they are shaped differently than those on single speed systems. So if you're making a 22t coaster brake cog serve as part of a two-cog derailleur system, you can make it shift better by reshaping the teeth on a bench grinder. Take the tips off most or all of the teeth, and angle some of them one way, some the other way. You don't have to do this to the smaller cog (but that may help too).

All that said, if I were setting up a bike with an FW hub and wanted more range, and was resolved to having two shifters (so a lot of double shifting) I'd convert the hub to five speeds. The five speed hub is more reliable. It's a simple conversion, only a few parts need to be changed, and they are not hard to get.