There is just one secret in this section (in the Japanese version at least) and it can be found just before you leave. The left most block on the ledge leading to the exit contains a Mighty Drink. However, of much more interest is that door in the upper left corner. It has two different opening mechanisms depending on which version you are playing.
Japanese version: To open the door in the upper left corner, you must collect the bombs in the order indicated on the map. As you are awarded the 10,000 point bomb collection bonus, the door will open. As you step inside the treasure room, you will notice another door to the left. In order to pass through this door, you must be at power level three (green) and crash through it. However, before you go, make sure the tens digit of your score is 3. Jump until the last two digits of your score are 20, and then activate the green power and jump once more to crash through the door. If you arrive in this next treasure room with a 3 in the tens digit of your score, the door in the upper left corner of this next treasure room will also open for you, granting you access to a third treasure room beyond.
American version: It's a little easier in this case, but not by much. To open the door in the upper left corner, you must find the Sphinx hidden at the bottom of this section. It is just below the bomb which is closest to the flames. In fact, it is so close to the flame that you have to float down as you're about to pick it up and immediately move to the right once you do. If you collect the Sphinx, the door in the upper left will open for you. The left door in the first treasure room opens the same way: you must be at level three power (green), but you don't have to worry about your score. Simply enter the second treasure room. To open the door to the final treasure room on the left, a Sphinx is hidden in the room. It happens to be in the very top right corner of the room. Just jump up there and you'll land on a hidden treasure chest which contains the Sphinx that opens the door on the opposite side.
This Stage now bears a striking resemblance to Stage 9, which you may very well have skipped if you collected the first crystal ball. You will get another opportunity to jump from cloud to cloud as you climb up and over to another entrance back into the pyramid. As with Stage 9, there are a few treasures buried in the blocks that lie outside of the pyramid. Collect them if you can, but don't risk your life for them.
As with Stage 9, you begin a long ascent up into the sky via clouds. You start at the bottom and must climb all the way to the top. You don't have any blocks to worry about, so just concentrate on collecting balloons and avoiding enemies. Scroll them off the screen if their activity becomes too hectic.
Once you make it to the top, you have a long journey back to the pyramid. Don't focus on the bombs if it's going to get you in trouble with enemies. Focus on staying alive. It can be very tempting to want to unearth that hidden extra coin in the pyramid blocks, but it's not worth it if it's going to cost you a life to collect it. The enemies are at their most aggressive in this stage and beyond, so getting Jack safely inside the pyramid should be the priority.