Best time to submit application?

So I know applications have been open for a while and people have already had first round interviews, but I'm a bit conflicted on when to submit my application. Honestly, I haven't networked a whole lot last year, and I have yet to connect with the few contacts that I made over the summer. Within the next week or two, a few firms are coming to campus for networking, coffee chats, etc.

Would it be better to submit my application now, and then hope to connect with people, call people I already know, etc. and hope that they can help push my application through, or should I wait until after these events to submit my application? I feel like posting earlier is more advantage generally, but it wouldn't allow me to meet more people first.

Also, how do I go about asking someone to push my resume through? I'm planning on calling the people I've met soon, but I'm not sure what to ask. Previously I had questions about their time at the firm, etc. but I'm not sure what to talk about besides small talk? Is there a way to hint about pushing my resume without coming off strongly?

Thanks!