The phrase "second round" refers to a second chance or another opportunity to do something, especially after a previous attempt or round of something. Full definition
By second round of vaccinations, your puppy has a lot of protection. (houstonpettalk.com)
Hard game, i would be over moon with draw but even if we lose, we have good chance to go in second round. (justarsenal.com)
This issue has been discussed for years in international meetings of budget experts and to date no country allows for second round affects in budget planning and policy discussion. (3dpolicy.ca)