to lay the warp, to weave
絍 centers on the foundational step of weaving: laying the warp threads lengthwise on a loom so that weaving can begin. The two given meanings are closely connected stages of the same textile process.
絍 combines 糸 (thread) with 任, which likely contributes the sound and may carry a sense of bearing or undertaking, fitting the idea of setting up the warp threads that bear the weaving.
The left side 糸 is thread, and the right side 任 looks like a person bearing a load. Imagine a weaver carefully laying long warp threads, bearing the weight of the future fabric.
For ジン, picture a weaver laying warp threads while drinking a gin and tonic: gin -> ジン, and the threads are set straight like the drink's clean taste.