I found this interesting blog article on how to remove the slack between the extra stitches cast on with the backwards loop and the garment stitches.
I’m confused tho - once you’ve gotten to step 7 isn’t the yarn then technically coming out of the second row of now the first cast-on stitch? In other words, it’s in the middle of the row. To continue to knit the garment stitches, would I have to cut my yarn? That feels like it would defeat the extra work done to make the cast-on edge neat 
The last row before the cast on is made up of full stitches (pale blue). The cast on row extends that row but the stitches are only e-loops (lavender and blue) not full sts. The next row is a row of full sts (green), 4 over cast on e-loops and then continuing over the full sts of the main body. So the section of knitting over the newly cast on sts is always going to be a tiny bit shorter than the main body but this difference is unnoticeable.
You don’t have to cut your yarn just continue knitting. There are sturdier cast ons like the knit cast on or cable cast on that may be preferable in this instance.
Oh hmm - wouldn’t that throw off the tension, if I don’t cut the yarn? I’ve attached a photo:
I knit up to the first sleeve, slipped the sleeve stitches onto waste yarn, then on my right needle I cast on the underarm stitches using the method described in the blog post. Like the blog shows in its diagram for step seven, I put all the underarm stitches onto the needle I cast them from. But now the yarn is in the middle of the now, now at the end; and I need to knit the garment stitches from the opposite side of the sleeve to connect the front and back of the sweater.
Don’t knit back on the cast on sts until you’ve completed a round.
Cast on one fewer stitches than required then join to the other side of the underarm (blue arrow). Knit around, do the same thing at the second underarm then knit to the first underarm.
At the join of the body to the newly cast on sts, pick up a loop on the left needle (equivalent to the dark blue stitch in Techknitter’s step #5) and knit it, knit across the other underarm sts and continue on to the second underarm. Do the same thing there.
Techknitter’s post is using newly cast on sts specifically at the end of row. The technique works mid-row but with the above modification.
I find this a really good method for underarm stitches. You cast on two fewer stitches using the backward loop method and then carry on. When you get back round to that section again, you’ll notice that the junction between the body and the newly cast on stitches is loose, and you simply use that loose running strand to create an extra stitch either side. This corrects your stitch count and gets rid of the slack! The structure is exactly the same as if you had cast on the right number of stitches at the start without the slack!
