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

Most will say upgrade the source first. I think YMMV, try the first PC in different components and have a listen.

This may be of interest...I was upgrading my PC's and when I installed a very good 8awg cable from the receptacle to power conditioner, the clarity improved plus the quality and depth of bass was significant.
So for me the biggest upgrade was from the mains. 

Glad you asked. There really is only one way to do this. Very important: Begin by unplugging the first power cord. Then plug in the second one. Some say its best to unplug both ends before connecting the new one. But I have found it works just as well to unplug and plug in one end, then repeat at the other end.

One more thing. You will hear people talk about directionality. This is not what they mean. Does not matter which end you plug or unplug first. 

Cantankerous, shi$, MC.. LOL

Sit in the corner.. Pointed hat ON, young man...

Geez.. LOL OP.. Don’t let him goat you into the tongue test..

"plug into the wall and touch your tongue to the other end"..Got to watch him.. LOL Just kidding..

The Wall first, you’ll hear the improvement downstream. You can’t if there is a problem upstream.. Just makes sense.. Then the Integrated.

Regards..

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.