Former U.S. Senator Larry Pressler is currently petitioning to get on the South Dakota ballot as an independent for U.S. Senate. Here is an article about his campaign, which has a reference to his petition drive. The deadline is April 29 but the primary (which he is not involved with) is not until June 3. Pressler probably doesn’t need extra time to get his 3,171 valid signatures. But if he did need more time, it is almost certain that he would win a lawsuit against the deadline. There are no legal precedents upholding an independent candidate petition deadline that is earlier than primary day.
The South Dakota non-presidential independent petition deadline had been in June until 2012, when it was moved to April.