What is the best order to upgrade a power cable loom? Thanks.

Potential problem with first getting just one power cord and trying it everywhere one by one is that different components may require different power cords. Unless, and even possibly then, you go with top of the line for all components.
I like and use Purist Audio and Wywires cords.
My best guess would be, yes, to start with the power cord from the wall. Question is which one ?
Wywires has a 30 day return policy and his cables and cords are on sale now. You could try Platinum High Current. Or Silver HC. I use Silver Digital with my DAC. I did try it from the wall to power regenerator. It was pretty good but Purist was better. For the DAC, however, the Wywires was better.

Top of the line PC?.. I make my own Top of the Line PC.. NEVER more than 60-70.00 in parts for a 1 meter. A good weave, conductors, covers, and terminal ends.. #10 Silver clad, mill spec with teflon covers. Armor and red copper terminal ends with carbon fiber terminal covers. A 4 wire Weave (3 silver clad (Low Strand count) and 1 copper MS), treat the wire ends and tighten the CRAP out of the fasteners OR solder or BOTH.. Throw them on the conditioner for 7-9 days for #10 silver clad. 8-10 days for pure silver multi strand. 10+ for solid core...

I NEVER found better. Prettier, Maybe.. Not much though, when I add cable armor, collars, and shrink tube.. :-)

Save your money do your own PC, Save some money.. BIG money..

Regards

different components may require different power cords.

Yes, I should have mentioned this. I'm using 14 gauge for low-current-draw components. Power amps which draw higher current may require 8, 9, or 10awg. My custom PC from the mains is 8awg with gold Furutech terminations. 

And then there are the different designs using copper strand, silver, and copper+silver. Definitely find a dealer with a return policy or trial period.

The moment someone start talking about “gauge” as the description of the cables they are using, as if that is a brand, I stop reading 😂😂

OK. Cullen, Furutech, VH Audio. I don't think this helps answer the OP's query.