After stopping not wanting to go home, Araragi Koyomi and Hanekawa Tsubasa keep seeing Hachikuji on the streets. Why does this still keep happening? I thought Hachikuji can only be seen by a person who doesn't want to go home.
Do the effects of the first encounter continue indefinitely, or is this the result of Hachikuji leveling up in the ghost ranks?
As you stated, Hachikuji had one "level UP" as Ghost.
This means she is no more a "Snail", but moreover a "Wandering Spirit".
This means she can generally be seen by anyone having the eyes to see.
Also she no more haunts people not wanting to return home, but strolls about.
In Onimonogatari:
It is revealed that Hachikuji's "promotion" was a "lie". As the darkness began pursuing her when she accompanied Araragi home, in direct contradiction of the nature of the mayoiushi, it is clear that she continued to possess it in some fashion--and immediately before that, when he offered to take her out to eat, she notes that he would seem "like those parents who order enough food for themselves and their late daughter", implying that to the best of her knowledge she remains invisible to the general populace.
In Hanekawa's case, every time she encountered Hachikuji, her living situation had not improved, so she simply retained the desire not to go home.
As for why Araragi can see her, Ononoki cites it as proving that he retained some level of vampirism, implying that it is generally the case that vampires can see ghosts. (Naturally, this would also explain why Shinobu can see her.)
It would be reasonable to assume that Ononoki also has some special capacity to see ghosts as an onmyouji's shikigami (and one created for such a purpose, no less), and Gaen may just perceive her through her powers of clairvoyance.
This does present one complication, namely that in Owarimonogatari (Ougi Dark arc):
Araragi could see her after having fully regained his humanity.
However, having been at this point forcibly extracted from Hell after passing on, it stands to reason that she no longer had the properties of her prior existence as a mayoiushi, and was instead some other type of ghost--the cause of her existence as a ghost no longer being the circumstances of her death. It's possible either that she could be seen unconditionally, or that Araragi was uniquely able to see her because he personally brought her back.
Although Gaen states that the darkness would swallow her if she did not become a god or return to Hell, this may be because such a type of ghost should not exist to begin with, or rather that it defies the nature of the human soul to return from the afterlife while still dead. Failing that, she may have embellished things somewhat.
