Your proposal can be done, legally, pretty much as you drew it - at least, I'm confident that you can have a subpanel of a subpanel.

That said, it's not an installation I've seen a lot of.

I'd be curious to know - and I guess I'm adding a question to your question - whether or not it is legal (I'm not asking if it's a good idea) to omit the 100A breaker in the "middle" panel (the outdoor one in your diagram)...

First thing you need to do is find out what NEC version your location uses. It's not always the latest. You can get a hard copy from most of the major real and online bookstores (even the public library often has a copy). I like the "NEC HANDBOOK" which is a complete copy of the NEC as well as some additional pictures and explanatory text.

Your picture is fine as it goes. Be aware that there are specific requirements for spa wiring.

A couple of comments on the proposed setup.
Why feed the spa subpanel from the main panel and have a passthrough to the garage panel? I would feed the spa subpanel from the garage sub. Likely you won't have enough cable to splice in a sub anyway between the house and garage. You can use the new sub as a junction box, but that's messy.
Your proposed setup might also require a 150A double pole breaker in the main panel and appropriate recabling if you intend to run the spa and all garage loads concurrently. However, it's doubtful that you would need that kind of ampacity. I think it would be simpler and cheaper to feed the spa sub from the garage sub with a DP 50A breaker.

That would presume that he actually has 100A of load in the garage.
I would guess that if he only has 200A service total, that he's not using 150A in the tub/garage.

I am out of space in my house panel...so, can't add a breaker there. I don't want to add a breaker in the garage because then I would have to dig a trench back to the hot tub and lay another 50-foot cable run in parallel to the current 100A feed. The hot tub is sitting right next to the spot where the cable to the garage exits the side of the house. However, adding a breaker to the garage and running it back to the hot-tub may be the "most legal" solution and would omit the need for the new sub-panel.

If I do go with the sub-panel approach, I am concerned about whether I need to upgrade the breaker in the house main panel to 150A. I don't plan to run heavy loads in the garage at the same time that the hot tub is running.

I plan to read the hot-tub specific section of the NEC before I start.

Thanks for all of your inputs! Please keep them coming! :)

The big issue is how you are going to going to insert that subpanel in the existing wiring. You're likely not going to have enough free wire to put it in directly, so you're going to have to add another junction or repull some of it.

By the time you get done it may have been easier/cheaper to dig that trench.

Another reason to go with the sub-panel is that NEC 680.12 states: "A maintenance disconnecting means is required for a permanently installed pool, storable pool, outdoor spa, outdoor hot tub, or fountain equipment, other than lighting for these water bodies. The maintenance disconnecting means must be readily accessible and located within sight from the permanently installed pool, storable pool, outdoor spa, outdoor hot tub, or fountain equipment."

If I place my GFCI breaker in the garage, then it's not within line-of-sight.

The cable run from the 100A breaker in the house is very short, so if I need to re-run a 10-foot cable so that I can put a sub-panel in there - so be it - as long as it's legally within the rules of the NEC to do so.

There has to be a GFCI, there has to be a maintenance disconnect. Nothing says they all have to be in the same place. My GFCI is in the main panel and have a separate disconnect box at the spa.