The halacha says that a woman is required to cover her hair. You are right that it is inferred from the parsha of sotah, but if you are familiar with the way halacha works, many things are not stated outright, but there is a system of drashos which the early rabbis used to derive laws from the Torah, and that system is part of the oral tradition which we believe was transmitted to Moshe along with the written Torah. Therefore, technically speaking, anything derived through that system will generally have the status of being derived directly from the Torah.
It is commonly said that women must cover their heads because it is not tznius, etc. I do not believe this to be so, or at least the primary reason. We don’t generally qualify laws derived from the Torah with reasons, because at the end of the day, since it wasn’t something enacted by the rabbis, any reason we give would be pure speculation, as God didn’t tell us his reason. What we can do, is look at the halacha from a legal perspective, figure out all of the technical details and loopholes, and then, after all that, we see the overall picture and think to ourselves, how can we find personal meaning in this halacha? But this has to come after the halacha is established with all its technicalities, not before, because how can we allow ourselves to be biased by a reason that is pure speculation?
Why some of the wives of great people didn’t cover their hair, is a good question. Maybe since everyone was doing it, the rabbis decided to pick their battles and just keep quiet about this. I don’t really have a definitive answer, but the halacha is still clear, from the Gemara down to the Shulchan Aruch, that women are required to cover their heads when they leave the house. Inside the house is a different story, and may quite possibly be permitted.
As an aside, I am not a rav and this is not a psak, but from my understanding of the halacha your method of covering your head is fine.