As has been pointed out, there are no "rules" stopping you from having a sequel starting within the time frame of another book in the series. "The Horse and His Boy" takes place wholly within the last few chapters of "The Lion, The Witch, and The Wardrobe", for example.
Thinking like a reader, if I loved a book and its characters and I read another book where I can see different characters tangentially intersecting with events I'm already aware of - I would LOVE it. Not only will I be reading a fresh new story with a great new character (or two) and different stakes but I will get a chance to briefly revisit ones I loved without the story directly continuing the last one.
Far from saying this is ill-advised, I would go so far as to say this is a brilliant idea. I wish more authors did things like this.
Advice for writing the sequel
Disclaimer: Your mileage may vary, this is just my opinion.
I would strongly recommend setting out early to anchor your readers with a sense of where and when the new story takes place within the wider fictional universe. That's not to say that an "ah-ha" moment where they realise halfway through couldn't work, just that it carries more risk. It really is up to you if you think you have the skills to pull a late reveal off or not.
Let us get attached to the new characters long before we meet anyone central to the previous story. The reason being is that readers may expect the established character to come in and take over. Meeting established support characters early is unlikely to carry that same impression.
Let us see how the events in the previous story ripple out into this story. I would caution against solving any problems for your characters with those ripples but making additional problems will probably make for a gripping read.
My final advice: in all things writing, if you wonder "should I do this?" or "am I even allowed?" do it anyway. Try stuff. Break rules. If it works, great; if it sucks, tuck it away and pretend it never happened (or figure out what's wrong and fix it). Whatever you write is not set in stone - so experiments are always entirely recommended.