The phrase "late push" refers to putting extra effort or trying harder towards the end of a task or event in order to achieve success or reach a goal. Full definition
The carrier announced earlier this week that the update would be released today, but it was later pushed back to its current scheduling. (theverge.com)
One example from my story is that I had a character sitting on a cot, then moments later he pushes himself up off of the floor. (lonetrout.com)
The developmental editor later pushed me to include even more than what I added on my own. (blog.publishedandprofitable.com)