Tools from Tetra with the (I think singular shared) item ID "tetra:modular_double" and tools from Tinkers Construct with IDs like "tconstruct:pickaxe", "tconstruct:hand_axe", etc don't go into the tool slots for automatic use from inventory, which is a genuine bummer right now without a configurable whitelist built into Inventorio for manual editing/fixing.
"tetra:modular_double" should probably be allowed in any slot given how wildly customizable they are.
tconstruct:pickaxe, tconstruct:sledge_hammer, tconstruct:vein_hammer really should be allowed in the pickaxe slot, tconstruct:mattock in either any hatchet, shovel, or hoe slot, tconstruct:excavator in the shovel slot, tconstruct:hand_axe and tconstruct:broad_axe in the hatchet slot, tconstruct:kama, tconstruct:sword, and tconstruct:cleaver in the sword slot, and tconstruct:scythe in the hoe slot. An argument could be made for tconstruct:sledge_hammer and tconstruct:broad_axe in the sword slot as well, but really just being able to use those as default tools in the belt as intended by Inventorio would be a really welcome enhancement.