You have to travel about 1000 LYs out and travel away from the "tourist routes" to scenic or interesting locations (like nebulae) to get a reasonable chance of finding Untagged or partially Untagged systems. But it only takes from a day to a week to get out that far (depending on your jump range and on how much of a rush you are in). Many players seeking Tags jump over this already-explored space as quickly as possible, to get out into fresh unexplored regions.
Sorry, but I think you accidentally made a mistake here, which could mislead a beginner, with your "a day to a week" comment. If we're talking about going 1,000 ly out - a good estimate for where you're likely to start seeing unexplored systems, assuming you headed in a less popular direction - then you can do that under an hour, even in an unengineered ship. Lots of ships can reach a 35 ly jump range, and even then, going that distance would take ~30 jumps, which would take half an hour if the Commander isn't stopping to scan anything.
If you're talking about going to Colonia, then with the help of neutron stars and a modded FSD, getting there could take a day to a week in real time, depending on how much time per day the player spends in-game. (Which is why I prefer to say "x hours of playtime" instead.)
Anyway, back to the original point: there's also that whenever there are new stations, they "pop into" existence, usually as a fully-fledged station, or as an under-construction model that has some services missing. There's no mechanic for NPCs constructing and over time finishing a station... yet? The addition of damaged and repairing stations in 2.4 (related to Thargoid attacks) offers some potential here, as the same mechanic could (and probably will) later be extended to allow players to help newly-constructed stations to grow. In my opinion, even just that would still be better than the usual "oh yeah, there's suddenly a fully built asteroid base near that nebula".
Also, as Major Klutz said, the EDSM image is misleading in this small size. If you check it in depth, you'll see that there are still plenty of areas in the galaxy where only a handful of Commanders have zipped through, and where you could fly and not see any other "lines" within even a thousand lightyears.