If you're looking to kill bugs from a distance, then you're going to need to upgrade your bow! While upgrading sounds good in theory, you're going to need to be able to slaughter some spiders to get the parts needed to create this thing in the first place. So, make sure you've got at least an upgraded melee weapon, so you can take out the bugs needed to craft this buffed up bow.
To get the tier 2 Insect Bow, you're going to need to kill some spiders to obtain Spider Chunks, Spider Fangs, and Spider Silk. Once you've collected these ingredients for the recipe, you can head back to your workbench and craft it.
The easy part of this is acquiring Spider Silk. Wolf Spiders don't appear to drop it, so you will need to go elsewhere to get it. Thankfully, it is fairly easy to farm because you can harvest it from webs or killing Spiderlings. The most reliable place I found for webs is to the southwest under the huge rake that is a big landmark on the map. If you look under the big boulder that is propping up the rake, you'll see the following cave with plenty of webs to harvest:
You can also find locations where you can hack Spider Silk near the big oak tree or anywhere you find spiders. However, you can only harvest the flat spider webs, you can't collect silk from the webs that the Orb Weavers make. They look triangular and when you run up to one, it won't say "Spider Web" like the other ones do.
Now that you have your silk, it's time to slay some spiders. We don't just need to kill any spider, we have to specifically target Wolf Spiders. Orb Weavers do not drop Spider Fangs, which are a requirement for the bow. There's a couple of ways to get some easy kills on spiders, so here's a video on one of them:
Depending on your luck, you might only need to kill one of them, but two should be the maximum you need to kill to get the fangs and the chunks. Once you've got the ingredients, head back to base and research the fangs and chunks to learn the recipe. Then all you will need to do is craft it and you will have yourself the tier 2 bow!