Trying to fix a large company rat nest network

Recently I got my first job that let me really stick my nose into the IT world and OHHHH boy I've been enjoying the hell out of it, but I've run into a problem that I'm really too new to understand how to fix it and I'm less intimidated and more enthusiastic to fix it. Even if I don't end up fixing it, learning how to do it in theory would really be enjoyable for me.

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So, my company has ran since basically the dawn of time. They're very slow at adding new technology and they don't always allow the budget or time to do things right. Because of this, we don't hire contractors for our wiring and we've just had facilities wire our extremely large building over time. This has caused a tremendous amount of problems and the largest network rats nest I've ever read about in my entire life.

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I was responsible for setting up a new department for the company, so I did everything from installing PC's, carrying them in, setting them up, setting IP's, and the other stuff, when one guy just starts ripping out all the cables from the switch! In my head I'm freaking the hell out, but he was my superior, so I assumed he knew best. Next thing you know, 1/2 the network goes out for the building and I'm up the rest of the night fixing what my supervisor messed up.

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But Here's the cool part. I got to play with the switch and hang around the engineers who had to come in and fix it! I learned a massive amount, but I was also surprisingly trusted (for some god forsaken reason) to set up the switch and run the cables, set up the network, and was even introduced to how to play with the switch through putty. I began to make a layout of the department and started tracing every port and wire. I began to learn that facilities didn't even trace wires to the closest switch and the rats nest was worse than I thought. Over the entire week, I've begun just tracing every port I run across and running my wand across the switches to learn where they lead, and I've begun to make a network topology of the entire building. It's a very slow process and I was wondering if you guys had any ideas that could help me speed up the process.

I've slowly begun to clean up the network too and organize, label, and document everything. One of the engineers has been helping me find ports I can't trace too because he's been incredibly encouraging and he also wishes to fix the rats nest. Is running my wand across every switch port by port really my only option? I'm horribly eager to learn, but getting the layout for this building could take me 2+ more months of work alone and I'm also tasked with other jobs, so I can't be doing this all day. I'm just really trying to fix this, but I'm kinda stumped as to how to speed it up.