I'm interested in professional mentors, and I'm curious as to how to leverage my network in this search, or if it's something that has to happen naturally.
I don't think you have to wait for it to happen naturally. I think you can definitely seek out a specific type of mentor. Now, something to keep in mind though is that seeking a mentor is inherently a selfish endeavor. Meaning, you're goal is to find someone who has already done something you respect and your goal is to take up their time to help you achieve the same thing. I say this because you should be cognizant that not everyone wants to take on a mentee because it's definitely work, and that also it's worth exploring how you can provide value to them and not just the other way around. That being said, seek out people who can help you with specific things or a specific trajectory. Don't just try to find a mentor and be like, "help me" – that's a prime example of a waste of their time. If you find the right person, show them specifically where you're looking for help, and try to create value for them as well, I think you can be very successful in seeking out a specific mentor!
I spend almost all of my time outside of class networking with professional mentors rather then doing other things. In my opinion I started off by going to networking events and over there that is something you have to look for. Professionals that come to a networking event come with the intention of finding a mentee at most times, or are at least open to the idea. I met my first mentor at a college networking event and he introduced me to his whole network of professionals. The first mentor is the hard part after that is its much more easier. I reccomend you go to networking events and meet people. There isn't a right or wrong way just don't be creepy and be genuine with your questions, and career plans.
There are a few different tips and tricks to finding a good mentor. First, I would look for someone who has similar beliefs or views as you to avoid undue tension. Second, I would look for someone well established in their field and career who has kids. Why kids do you ask well it is simple really parents tend to be more lenient in the learning process and encourage growth. Finally, look for someone who is successful yet lives beneath their means.