An S-1 registration statement is typically required when a company is:
Going public (initial public offering, IPO)
Registering new securities for sale to the public
Performing certain types of secondary offerings
When an S-1 is not required:
If a company simply wants to:
Change its ticker symbol
Rebrand or rename itself
Move from one exchange to another (e.g. NASDAQ to NYSE)
…it does not need to file an S-1.
What is required for a ticker change?
Notify the stock exchange (NASDAQ, NYSE, etc.) and follow their procedures
File an 8-K with the SEC if the ticker change is part of a broader material corporate event (e.g. a name change)
Update public disclosures and investor communications
A ticker change is a procedural/exchange-level matter, not a securities registration event. An S-1 would only come into play if the ticker change were part of an IPO, uplisting with new shares, or other public offering.