Reminder that Ichigo is the Jor?gumo:
In the bathroom scene where Hiro washes his face, the butterfly that he notices is trapped and cannot fly. Since episode 1 and all throughout each subsequent one, we know that the ability to "fly" (to ride a FranXX) is what separates the "living" from the "dead" in DITF's world, or at least the world as seen through Hiro's eyes. In Hiro's first scene in the anime, for example, he notices a bird that flies into the canopy of the nest, injures itself and dies. That is part of the same motif.
It stands to reason then, that the spider's web represents the "bird cage" (that is what, ironically, the parasites call their home), which Hiro needs to escape. From episode 1 through episode 3, Hiro grows increasingly aware that he is living in a trap, episode 3 ending with his realisation that riding Strelizia together with 002 is his only chance to escape. Strelizia is the only FranXX he can use, and Hiro is the only stamen with whom 002 can ride without killing it in the process. Episode 3 also revealed that 002 wants to escape from the plantation(s) which she compares to graves, something that reinforces the association of the spider's web with the "bird's cage".