Does the Bible say anything about there being a second chance for salvation after the time of rapture? I would assume that God is doing this rapture so that people would recognize Him and repent. But I don't have a Biblical basis for my assumptions.
For starters, the Bible doesn't contain the word "rapture" nor does it clearly state that such an event will happen. Different people have read between the lines to infer a rapture at different points in the book of Revelation, but it is not clearly stated anywhere.
Since the Bible doesn't clearly state that there will be a Rapture and those who believe that it will happen disagree significantly on when it will happen, the only clear answer that can be given to your question is:
"No, the Bible does not make a clear statement about there being a second chance for salvation after the time of rapture."
If the Rapture takes place in Revelation 7:9-17 according to the Pre-wrth Rapture viewpoint, then there will be no further salvation. The Pre-wrath Rapture takes away the remaining saved; what's left is God's wrath in the trumpet and the bowl judgments. (That's why it's called the "Pre-wrath" Rapture.)
During those judgments, it never tells us of people repenting. Rather, they "will desire to die" (9:6); "they did not repent" (9:21); "blasphemed the name of God" (16:9); "blasphemed the God of heaven" (16:11); and gathered "them to the battle of that great day of God Almighty" (16:14). So the Bible gives us a dim view of the possibility of salvation when He pours out His wrath. What's left of humanity after the Rapture will receive His "true and righteous ... judgments" (16:7).
However it does state that we will be caught up with Him and He will return (1 Thes. 4:17).
Also if there was a second chance He would have to go back to the beginning of time and give everyone a second chance.
Now do you think that he will allow those who chose Hell for not believing in Christ to change their mind? No He won't you make your choice while you are here and when He does return it will be too late to make that choice