I think a step SW with the warrior might be handy (just to expose as much land as possible for the settler view). Then either SE to the jungle or SW-SE if that's flat land. After that heading E in general to scout out the land south of the capital.
The pretty pins are a feature of CQUI. The base game ones have far less useful symbols.
I'm also not convinced by the city site. Hopefully something will show up in the fog. Having the stone in first or second ring would definitely be an advantage.
I understand that it is possible to harvest straight into the thing you want to produce, you just need to have the queue empty at the time. The harvest income goes into the overflow. This doesn't matter here: I intend to harvest into an almost completed slinger with Agoge policy in place. That gives the entire harvest a +50% modifier, which should mean the overflow provides almost the full cost of the district.
Now some longer thoughts on how quickly things can be completed:
There are three limits on improving the dyes.
Production: 6.3

per turn, goes up to 8.4

from turn 8.
Slinger (35

) is done on turn 6, builder (50

) on turn 12. Sheep pasture on turn 13, wheat farm on turn 15, dyes plantation on turn 18.
Culture: 1.3

per turn, goes up to 1.5

from turn 8.
Needs three tiles at approximate costs of 10 + 20 + 35

. That's *way* in the future, so I'll probably have to look at buying a tile. I think the second tile will be claimed at about turn 22. I don't think the culture cost changes when you buy tiles for gold, so buying the wheat ASAP would minimise the time. Will have 50

on hand very soon to do this.
Research: 2.8

per turn, goes up to 3.4

from turn 8.
Three techs required: Animal Husbandry (25

), Pottery (25

), Irrigation (50

). Irrigation will be boosted by the wheat farm, so 75

total. I reckon that will be done at about turn 23.
So to take this route would cost 50 gold and get the inspiration on turn 23. Not at all convinced this is a good use of gold just for a dye tile. The only advantage of the dye is that it would jump me up from third to first choice of pantheon, barring religious city states.
Instead I think it would be better to push the settler up to third in the queue, getting the second city down quickly and improving a tile there. I believe that settler would be produced around turn 19 or 20. This plan is probably 5 or 6 turns slower as I'd be waiting for turn 22 to improve the wheat.
This is easy for barbs to disrupt, of course. Even so, I'd rather produce the builder before another slinger, if only because the pasture will produce an extra

. Not the end of the world if the settler has to be pushed back for extra military, the inspiration for craftsmanship can wait a while if it must. Good to get ASAP though, just for the excellent production cards.