I've had this question a couple of times in recent weeks, plus I've dealt with it with Bang2writers more than once, so I thought it could do with a post of its own.
The short answer is: YES. Of course you can. You can do anything you want. That's Spec Writing 101.
But wait, wait...
Is it a GOOD idea to write a sequel of your spec?
Hmmmm.... Not really. And I'll explain why.
We all have limited time, whether we're writers, readers or BigAss Execs. Potentially, writing a spec that DEPENDS on someone having read the FIRST one may not be the greatest idea. What if they didn't like the first one? Or what if they don't have time to read two scripts? It might severely impact on whether either even gets out the envelope or email inbox.
"But this story/these characters have so much more mileage!" writers complain.
Great. Everybody loves mileage. So here's what you do...
... Don't make that second spec a sequel in the traditional sense - MAKE IT A COMPANION INSTEAD.
In other words, have the same characters, the follow up story, WHATEVER --- but just make sure it STANDS ALONE.
Now you have the best of both worlds: you get to write the characters you love again... But that reader, producer, director, actor or whomever you're sending it to doesn't HAVE to read the first script in order to "get it".
You really do get 2 for the price of 1.
Of course you could do exactly the same for a prequel... Or for a WHOLE FRANCHISE. If your idea really will run and run, why not? If it's the best idea in the world - and who says it can't be? - I don't really see a producer somewhere down the line saying, "Oh you know what? We *were* interested in this idea until I saw you'd written thirteen individual scripts all with their own arc and and great merchandise potential". AS IF. Worst case scenario is, they'll take the one they like and ignore the rest. So go for it if that's what you want! No rules, remember.