Short answers: Get cited. Present. Show yourself to be a peer.
I have the opposite view about networking: you should not waste any day on it unless it directly helps your research (e.g., potential collaborations, working on grants). While admittedly you need to network and get yourself known, that won't help at all unless you're known for great work. You're not going to be known for great work without having great publications. They don't need to be in the best journals (though it helps), but you need to be pushing out work that answers questions that other scholars have. In my opinion, there is no better way to get your name out there than to be cited.
The single most important thing that I learned from transitioning out of my PhD program is this: Publishing is networking. Publishing is talking with your peers. (Particularly if you publish in conferences, you are literally talking to your peers.) No level of pavement-beating will get you more attention than to win a best paper award or to present on a topic that big people have been looking for a good citation on for years. I know a few relatively young scholars in my field who have just about universal name recognition. Their MO? 1. They answered questions that researchers were interested in knowing the answers to. or 2. They answered questions that people didn't even realize needed asking, but were so important everyone needs to talk about them now.
In the long run, being known doesn't help you much if you're known as "another one of those people who works on topic X." You want to be known as "An expert in topic Y" or "The first person I would go to with a question about Y." If you don't have this, any major researcher will say: "Why would I bother collaborating or recommending this guy?" Obviously, this takes years, but so does a doctorate. Nothing is more important to being well-known than asking the right question at the right time (and then answering it).
Additionally, one secondary route of networking not noted by others from what I have seen: acting as a point of contact who interacts with sponsoring organizations. You know what another great way to get face-time with experts is? Being one member of a small workshop or meeting for people all sponsored by the same funding agency. Usually this responsibility would be taken by your lab head or similar PI. To get this responsibility, you need to basically be the best grad student working with the PI and the PI needs to be unavailable or need a second pair of hands. However, by being there, it says two things: 1. You are the best student in the PI's lab/project and 2. Your PI trusts you enough to act as their proxy. I think this only works with an established (full prof) PI, as non-tenured PI's may need the networking as much or more than you. Networking with program officers is also a good habit to be in. Ultimately, these are the people who control what kinds of work can get funded. This is one of the ways a good advisor can plug you into their network (along with making sure to introduce you to their big-shot friends/rivals).