If you save up a scream in dual form and jump up there as pink (since you can't spin while clinging to change forms), when it zooms out to do it you can see that there are actually two rukhs there. You can also see that a parrot flies right into the nook from way over to the left near the Sun Temple exit. However, you cannot cling to parrots. You can't bind parrots and get pulled along by them. You can't like, springboard off of parrots so far as I can tell. It's definitely too far to soup-jump from the left, as was mentioned.
Briar-boosting (planting one of those spiky phallic things directly below yourself, which then launches you away from the point of planting) doesn't work for getting there from below, as there aren't any flat places to do it; they all face outward. If you could jump at acute angles in plant form off of briar rods, instead of only normal to their surface, it could be done by putting a rod sticking down and to the left from the middle ledge, and putting another rod perpendicular to it lower down on the lower ledge, and jumping up on those. But for some reason nature form got the short straw for jumping capability.
As an aside, I completely forgot the fourth, highest level that caps off the chimney and ends on the right at the edge of the sky. I can get tantalizingly close to getting on there by briar-boosting off of the one flat part of the highest chimney level, between the two nests. Anyone had any luck getting up there?
(Is the ruined submarine at the entrance to this part of the veil anything more than scenery?)
It looks like there's a fish form passage just above the bottom level in the chimney, on the upper left corner of that T intersection for the bottom level. But that wouldn't make any (useful) sense, so it probably isn't.
Update: You can briar-boost from the top (third) level of the chimney to the left side of the Sun Temple exit; it follows that you should be able to do the same in the other direction, and thus reach the middle (second) chimney level. Applying this theory has proved difficult. More results to follow.