The best way to add dimension to your plot is to every so often, pull out important plot elements at random, and ask yourself how they might be related, or how they might combine into a larger element. It won't work for every pairing, but sometimes just free thinking about what they have in common will spark serious inspiration.
Also, with very little work beforehand you can add the illusion of massive depth with the secrets deck. The secrets deck is a stack of cards, one for each major and minor plot element, character, location, item etc. Each card contains the name of the plot element, a little reminder of what it is and where it belongs, and a secret about it.
Every session, draw a few cards from the secrets deck, and somehow let one or two secrets fall into your players hands. They might ignore them, and not take the bait, and that's fine. Later on, when they're using the plot element extensively, produce the secret and slap them in the face with it. It will look like you foreshadowed this major thing for them as a tease knowing full well the adventure was going to get to where it is. PRO DM.
If they pursue the secret, you have a nifty little side adventure that you can run with, and it won't be hard to tie the sideline back into the main story.
Add to the secrets deck every time the characters discover something they are even the slightest bit interested in.
This works especially well with possessions that the characters treasure, because it makes them feel like they are central to the plot.
Edit: Articles about secrets