Asking someone in the final throes of the book production process for advice on academic publishing is asking for trouble. My non-serious reply would be “Run! Flee for your life!” But, slightly more helpfully, I would recommend acclimating yourself to the peculiar features of academic publishing, beginning with its specific temporality. Who knew what waiting a year to hear back from a press as to whether they were even interested in pursuing a manuscript would feel like? You won’t wait that long to hear back on an article submission, most likely, but it will still feel unreasonably long. Use professional meetings to meet editors, introduce your work to them, and (judiciously) gauge their interest level. If someone is expecting, or even better, has solicited, a submission, it will almost certainly get expedited. Another peculiarity to be aware of is the difference between the writing that will advance your career and the writing that just happens to need to get done and that someone wants to assign to a graduate student because you are around and presumably anxious to please. A lot of the most tempting opportunities for entry-level publication (I am thinking of encyclopedia entries, for instance, and to a lesser extent book reviews) count for very little on a CV.
A single substantive essay is a much better goal to set for yourself. In addition to the collectives Ashley Dawson mentions, consider collaborations: not just co-writing an essay (if that feels too difficult) but co-editing a section or a special issue of a journal. Many journals, and not just graduate student one, are open to such proposals from you. Learn the conventions of academic prose by editing others, and make contacts in the field at the same time. And finally, don’t lose sight of the importance of quality over quantity, particularly at the early stage of one’s career. I feel guilty giving this advice knowing how much academia prizes publication, even for its own sake, a situation I don’t believe will change even as opportunities for publication, particularly in the traditional form of physical books and journals, continue to narrow. But I do think the emphasis on professionalism can be corrosive if it has graduate students scrambling to present and publish their work from the very first years of graduate school. A good time to publish, at least in my field, is once you are at All But Dissertation status. And again, a single article in a recognized journal in your field will take you further than any other combination of activities I can think of. And don’t forget to make use of your mentors, who should be able to tell you when your work is ready and where might be a good place to try. If they are unavailable, unhelpful, or obtuse (none of which is rare, sadly), get other ones. Don’t stop trying until you get the answers and feedback you need.