I'm no further on in my academic career ambitions than you are (so please take anything I say with a grain of salt or two!), but I've thought about it quite a bit. Some things you could do, at this point, for academia-related CV-building include:
-Presenting your work at conferences. (Much easier to achieve than a respectable publication, and can be very interesting.)
-Publishing in venues which, while not anonymously peer-reviewed, can serve as testing grounds for early-career people - the Oxonian Review is a good example. (I've seen junior academics include OR pieces on lists of their publications.)
-Getting involved with interdisciplinary graduate things at college level, if anything if that sort's available to you - I'm thinking of, for instance, graduate colloquia or a grad journal.
-Being a graduate representative on relevant faculty committees.
-Applying for any relevant awards or grants.
Of course I am not saying for one moment that any of these things guarantees success - I know from friends' accounts that it's possible to do all this and still struggle to find employment. Needless to say, doing good work is the most important thing! But I think that seeking out opportunities for professional development could potentially give you a little extra push (and some of these examples, e.g. conference-going, might also directly improve your work and lead to earlier publication).
(There are, actually, a couple of people on this forum who published very early - and I know a couple more in real life. But most people have assured me that early prestige publications are rare indeed. I've done some of the things on my list above, but I'm aware at the same time that I'm just not ready to submit to major journals.)