Math PhD advice?

This past winter I realized I had enough credits to graduate a year early from undergrad, so I hastily put together some applications to PhD programs. However, I wasn’t accepted into any of the programs I applied to, so now I’m taking a fourth year of undergrad to take some more classes, do more research, and improve my applications for next time. I’m definitely going to take the GRE and Math GRE this time around (the schools I applied to last time didn’t require either). I also am about to submit a paper I’ve been working on with a professor for publication, so I’ll be able to put that on my CV. Does anyone have any math specific (or general) advice so I can make sure I get an acceptance this time around?