You want another take on how a new series could be a semi-continuation of the old one?
A girl, about five years old, is playing in her back yard. Suddenly a vortex opens. Frightened, she hides in the bushes as four people come flying out. She can't see their faces but she hears them talking about sliding and interdimensional travel. Fans of the old show recognize the voices as Quinn, Wade, Rembrandt, and the professor. A moment later, one of them activates the timer and they slide out.
The girl comes out of the bushes and looks around, but sees nothing. She has an intense look on her face, as if she's found her life's passion despite her young age.
...
20 years later the girl is now a woman working in a basement lab much like Quinn's from the pilot. A man about her age enters the lab.
"What's so urgent, Quinn?"
"Wade! I've got it this time." She's standing in front of a white board with an equation indecipherable to 99% of the world. The equation is not complete.
"You've been saying that since we were in middle school."
"This time the equation is right." She adds the final term and punctuates it with a heart shaped smiley face.
The man spends a long time looking at the equation as she keeps talking.
"It's really simple, Wade, and it takes very little power. I can create an Einstein-Rosen-Podolsky bridge with a slightly modified cell phone." She holds up her phone, which has two small metal spikes sticking out of the top
The mention of a phone jolts Wade into action. "Other people need to see this. It has to be verified by someone smarter than me." He pulls out his phone and takes a picture of the equation. "I'm putting this on Facebook and Instagram."
She's miffed at his lack of faith in her. "It's right. Watch." She activates the app on her phone and a vortex appears in the middle of the room. "That's what I saw when I was five. That's the doorway to another world."
Wade is even more dumbfounded by this than he was by the equation. He starts to take a picture of this as well but Quinn stops him. "I'm not ready to share this yet. We have to test it first."
The episode proceeds with the new Quinn and Wade going to a world very much like Elvis World in the old pilot, except it's Michael Jackson in concert instead of Elvis and the radio guy talks about how "This country will never have a black president". When they come back the professor is at Quinn's door wanting to talk about the equation, having seen it in Wade's picture. Quinn shows off her discovery, they slide but Quinn adds too much power to accommodate the extra person and a fourth, who was just passing by, is caught in the vortex as well.
Where do they end up? Dunno, it could be anywhere...