Hi, Pam. Sorry you have to be here, but welcome.
Getting the first treatment 3 days after diagnosis speaks well of your oncologist. SCLC moves fast, so quick treatment is essential.
I don't think you "have" to have a second opinion at this point. See how the next scan goes, and how the dr. handles things in the meantime. Find out the credentials and connections your dr. has, if you haven't already. How much experience does he have with sclc? Do you like him, feel like he has a positive plan?
My mom, Bellringer, has not had a second opinion, but her onc. is in contact with other facilities and oncologists. In a way, it feels like she gets second opinions because of how her onc. works.