Reddit mentions of Sure-Max 12 Heavy-Duty Moving & Packing Blankets - Ultra Thick Pro - 80" x 72" (65 lb/dz weight) - Professional Quilted Shipping Furniture Pads Black

These are going to be a few different ideas for you to think about.

First of all, temporary "soundproofing" that you can remove from a rental property and take to another is nearly impossible.

Second, if you're hearing planes and trains and they are not too close, the energy you're trying to stop may not be those troublesome and hard-to-control extreme low frequencies.

Third, you might want to spend your money on the right microphone and recording technique instead of a lot on physical booth. This, of course, depends on whether your clients ever see the peculiar setup you are using.

So here's what I'm suggesting:

Continue to use your closet as your booth. You need to surround yourself with some density of sound absorbing materials. The smaller the space (and I presume you have a small closet) the deader you want it to be. Obviously treating a small space is easier than treating a big space, but to make it sound good you have to make it dead. Very dead. So buy yourself a 12-pack of these and if that's not enough get another 12-pack. Start by putting a couple of them on the floor, as thick as you can get them and still not trip, then more on the walls and ceiling. Fold 'em just right and you can cover the door, the make a flap to cover the door again on the inside. You should have three or four thicknesses everywhere, until you are sure it's quiet enough. I'll guarantee this will work as well or better than any expensive portable booth. Forget about the little foam backgrounds for your mic. They really don't work.

Next, get yourself a primo microphone. If you can possibly afford it, buy yourself a Neumann U-87, pretty much the gold standard. You are the artist, get the right instrument that will be good for a lifetime. (I have a Neumann U-67 I bought new in 1964 and it is still as good as ever and I have never needed another voice-over microphone.) One main reason the U-87 is a great mic for your application is its amazingly clean figure-eight pattern. By using the deep "nulls" on the side, you can reject sources of noise that a cardiod pattern can't. The built-in low-cut filter is very effective in getting rid of sub-sonic noise without hurting voice quality. It also has a nice proximity effect that can gently flatter your voice.

Now go in that booth and put the mic at a 45º angle to your mouth and with the back side of the mic aimed into the densest part of your blankets. Don't use a pop filter. It isn't necessary if you use the mic correctly, and it does take a tiny bit of the edge off the very top-end. Work the mic at 8-12 inches, closer (but still at 45º) to get that movie trailer sound, a little farther away for a natural, airy sound. Main rule: always aim the mic at your mouth, never aim your mouth at the mic.

Third, record (or process later) with an expander. Clients are probably going to be editing your voice-over tracks and they expect near-total silence between sentences. They don't want to have to add room tone to pauses because your room wasn't quiet enough. The expander should just "crush the silence" so there's no room rumble. Clients will think you have a big, quiet studio. Use as low a threshold as possible to eliminate any residual noise. Don't go by the meter here. Get the noise low enough so you can't hear it at normal volume levels.

Hope this helps:

Me: over 50 years of doing commercials, designing and building studios, and trying to figure out easy ways to do things.