One way to squeeze in some secret rooms in a round tower is not to have floors. Like, how do I explain this, Usually, on each floor you have a number of rooms, then stairs up to the next room, right?

Instead, you have rooms spiralling up through the tower, with each room connecting to the one before. So let's say you have a set of 4 rooms that together go through 360 degrees around the tower. There 4 rooms are connected by stairs, each room raised maybe 10 feet above the last one. They also go around a spiral staircase that reaches top to bottom, but only connects to the top and bottom of each set of 4 rooms. 

Now ... offset this slightly, and there's space for secret rooms here and there. 

I'm sure this makes no sense what so ever in writing, but I can picture it clearly. Has a certain elegance to it. Whichs is unlike me, I'm usually rather blunt and utilitarian. Which is a good thing, usually. 

I follow you perfectly.

Right now I plan to manage it this way ...

There are seven floors for the Wizard and his family above ground.  The "stairway" spirals in another cylinder next to the tower.  Every other floor there is a balcony that wraps around the tower, and this is used by the guards for their patrols, so it appears to the guards that there are four floors, not seven.  Each balcony also coincides with a doorway that leads to the tower interior, reinforcing the idea there are four floors.  If you go inside these doors, the ceiling appears to be high enough to reach to the floor of the room above, but the ceiling is an illusion.  The floors are spaced about twelve feet apart, but from the inside, with the illusion they appear to be about 24 feet apart.

Inside the tower, if you know where to go and what to do, you can access a secret stairway to the floor above and the floor below.  These are the mystery floors which are connected to the visible floors.  Once you walk up about six feet of stairs you can't be seen from the regular floor.  On these secret floors the wizard has his library, his lab and his experimental chamber.  The uppermost floor is his observatory where he can view the stars and anything else he wants to see.  This floor is circled with a ring of eight columns supporting an eight sided ring.  In foul weather the wizard can magically call upon forces to cover the ring and shield him from rain and high winds, but he can still see through this magical covering.  He usually leaves it deactivated because he enjoys the soft breeze.  He can even cause it to deliver shade from the brightest sun, enough that he could even observe the sun as through a deep smoky lens.

The "real" rooms have windows but they are covered with an illusion so the guards can't peak in.  They just look like stonework from the outside.

The Four real rooms from the bottom are the "living room", then the kitchen, then the bedrooms and finally the observatory.

The first basement level is storage for the wizard and is used for mundane food stuffs, blankets, out-of-season clothes and stuff like that.

The lowest basement is very deep with a high ceiling.  This is a place for his dragon friend to come and go.  His dragon friend arrives, transforms into human form, visits and then goes down to the basement to disappear.