You can modify your ships without having any of the shipbuilding stuff I think. You are limited, but you can add cargo space with some penalty to range and mass to help ease it that. Additionally, storage via outpost is cheap. It's like 3 iron, 2 aluminum, 2 adaptive frames for 250 mass resource storage. Build a couple of those at an outpost and you're set. If you do a....I don't remember the name, but a storage link between your ship and containers, you can transfer straight from ship to container without taking it out of cargo. Just mass dump things into storage and be cleared out.
Alternatively, if you have a lot of credits, Shieldbreaker, a class B ship at New Atlantis, is a wonderful ship. Like 2300 mass stock, and you can add more if needed with minimal penalty.