I agree that we should've some way of getting silk without spiders. Of course, it would be comparatively lower quality/value compared to spider silk.
Perhaps in the interim it could be a tree mod?
Mulberry Tree
Give Mulberry Tree a growth called "Cocoon", which can be harvested and processed into silk?
Well, if you're going to do it by modding, why not do it right?
It's not that hard ( just time consuming ) to slap in a silkworm ( creature ), mulberry tree ( harvestable for leaves ), sericulture building, and string processing for the realistic silk (material).
A very simple solution would be to add another hiveable insect: silk moths. They form colonies, your beekeeper harvests them, adds them to hives and the products can be refined. Usually by boiling the cocoons (kitchen?) and then spinning the thread (loom?)
In fact, multiple mods already use this system. Another solution is to make larger insect creatures and use their eggs to make silk, Kazoo had that great idea. Added silk materials to the eggs, accept them as reagent. Now your dwarves have to choose between using the eggs as food, for breeding or as silk source.
... Sorta like Meph said.