And even if you only have a Commercial Pilot certificate, you would still need those extended Second Class privileges to exercise your Commercial privileges from now through the end of June.
Boomer At AOPA:
Ron is spot on in his response. Here's the link to the FAA's Enforcement Policy: https://www.faa.gov/news/media/Enforcement_Policy_signed.pdf . Since your First class privileges would have expired as of March 31st, the Enforcement Policy applies to you.
That Enforcement Policy has been overtaken by SFAR 118. There's no issue of enforcement any more since all medicals expiring 3/31 through 5/31 are now legally extended to 6/30. For those who haven't seen that SFAR...
Of course, if your medical expired 2/28, it remains expired, and that enforcement policy (or more accurately, no-enforcement policy) doesn't apply.