After making an Instructable on recycling magazines into stomp rockets last year, I continued experimenting and here is my improved safer version with better launcher and better nose cones.

Supplies:

glossy magazine with fairly light paper (e.g., This Old House)
paper glue
gorilla glue
Optional: superglue or PVC cement
Optional: wide packing tape
scrap styrofoam packing material (finer grained is better)
5-6 feet of 1/2" PVC pipe
1/2" CPVC wing elbow (e.g., this one)
16 oz water bottle or 2L soda bottle
Optional: glow in the dark paint

Tools:

scissors
box cutters
drill with 1/2" drill bit (Forstner works very well for this)
flat file (nail file should do)

Step 1: Rocket: Fuselage

Cut a full sheet of paper from magazine. Put paper glue along one long side, avoiding excess but spreading a thin layer of glue up to the edge.

Wrap page around a piece of PVC pipe, starting with the glue side, glue facing out.

Make sure the resulting paper tube is loose enough to slide smoothly on the tube, but no looser than that. This takes some practice as the glue will grab the paper after the first loop. You might want to make the first few rockets without the line of inner glue for practice. (Experience shows that the rockets with the line of inner glue are more durable.)

Once the wrapping is complete, glue down the loose flap.

Optional: Put a layer of packing tape on the top four inches of the fuselage to prevent damage on landing. The top half of the fuselage otherwise has a tendency to crumple.

Step 2: Rocket: Fins

Fold a mail-in insert from the magazine, or some other piece of paper of similar weight, four ways and cut out fins, making sure there is a foldable flap for attaching them to the fuselage.

If you can judge by eye how to make the fins be evenly spaced around the fuselage, that's great. If you can't, fold the bottom of the fuselage flat to get two fold marks 180 degrees apart, unfold, and then fold again with the initial fold marks at the sides to get two more fold marks, and use the fold marks for spacing.

Insert PVC tube in rocket and glue on the fins (the tube provides support as you glue in the fins).

Step 3: Rocket: Nose Cone

Use box cutters to cut a piece of styrofoam that's about 1" long with a square cross section whose size is slightly larger (maybe something like 1/8" or a bit less) than the diameter of your rocket fuselage. Gently sawing back and forth with the box cutters makes for a smoother cut.

Use box cutters, file, sandpaper and/or hand pressure to shape the piece into a cylinder slightly larger than the rocket fuselage. This time, I used a file and hand pressure. I recommend wearing a breathing mask because of the dust. This is messy.

Round off one end. (For subsonic flight, a rounded nose is more aerodynamic than sharp ends.)

Squeeze the wide end of the nose cone until you can push it into the rocket tube with half of the nose cone sticking out.

Put Gorilla glue on the part of the nose cone that will sit inside the tube and install nose cone in tube. I recommend using gloves or else there is no avoiding getting Gorilla glue on your hands, and it's a nuisance to clean off. Wipe off excess glue carefully as it will foam up.

Glow in the dark paint on the cone and fins makes for fun with night launches. Use an ultraviolet LED to charge.

Step 4: Launcher: Tube

Cut PVC pipe into two pieces. I had one piece (eventually horizontal) that was about 3.5 feet and another (eventually vertical) that was about 1.5 feet. Glue the two halves into the wing elbow. (I used superglue, though PVC cement would be better. Haven't tried Gorilla glue. The pressure shouldn't be very large because the designs are not meant to be airtight.) Get a piece of scrap wood and screw the elbow onto it.

Chamfer (i.e., round off) the vertical end that the rocket will go on to reduce damage to rockets from slipping them on.

Step 5: Launcher: Bottle

Background: While everybody seems to power the launcher with a 2L soda bottle, I found a 16 oz (500 mL) water bottle to be more durable and to work better at least as compared to a single-foot launch (a 2L bottle might be better for a two-foot stomp). I am guessing that part of the reason the 2L soda bottle doesn't work better is that it takes longer to compress and may not compress completely, and so a lot of the air in it may be wasted. On the other hand, the small water bottle compresses quickly and completely. I also have not found any noticeable benefit to sealing the connection between the PVC pipe and the bottle thoroughly. The simple connection I use makes it easy to inflate and swap bottles.

Instructions: Choose a water bottle made of thin flexible plastic. (The cheap HEB water bottles work well for us here in Texas.) Drill a 1/2" hole in the cap. I drilled very slowly and carefully so that the drill would be unlikely to slip onto the hand holding the bottle, and if it did, it wouldn't do much damage.

When I used my cheap Harbor Freight Forstner bit, the resulting hole was just right, making a good seal with the pipe.

Step 6: Launching!

Put cap on bottle, insert a few inches of tubing through the hole in the cap, put a rocket on the other end, wear safety glasses, and ensure no one is hovering over the rocket. Then stomp!

The foam noses make the rockets survive launching in the front hallway of our house, where they have two stories available to go up before they hit the ceiling with the cone, bounce off, go down nose-first, and bounce off the nose cone on landing.

To re-inflate bottle, just slide it right off the tube and blow into it (another Instructables author suggests that for hygiene one can blow through one's hand--a great idea). When one bottle wears out, another should be at hand. The quick slide of the bottle makes for much faster launching than a screw-on setup.

Eventually, the hole in the cap may get too large and drilling a new cap might be a good idea. Usually, a couple of bottles will wear out before the cap does.

Preliminary measurements:

My quick measurements using my open source Two Point Height app (in the mode where you sight along the side of the phone) suggest a launch height of 60-75 feet. I read about someone getting 100 feet with a 2L soda bottle and two feet, so maybe you can do better, but the convenience of the water bottles is hard to beat.

I have been trying to measure speed by having a fin (on a model where the fins have more uniform height) intersect the beam of a red laser pointer shining at an infrared LED (LEDs also function as weak photocells; an infrared LED nicely responds to red light) hooked up to an oscilloscope, but the results are somewhat odd--the beam interruption seems more gradual than it should be. I was thinking I'm getting around 60 feet per second with a bare foot indoor stomp and a worn bottle. More data is needed.