You can resaw your 2×6 on the table saw. Raise your blade as high as it will go and saw through as far as you can. Turn it over and saw the other half. If it does not go all the way through, finish the cut with a hand saw and then plane or sand to get it smooth.
You lose a little more material, but other than that, it works well.
Get yourself a thin kerf rip blade which will take less power to turn the blade and use a splitter or a wedge to keep the two cut pieces from binding and one other thing is if you have a bow in the 2×6 keep it facing the fence or you are going to wind up with thin spots and watch out for kick back.
For making “deeping” cuts like this in a board you can do as suggested and do it on the tablesaw. The problem with that is you will often need to remove the riving knife so by the very nature of the cut your trying to make the initial cut will close down which can cause a kick back. It’s more likely though to increase the friction on the blade potentially causing distortion and certainly creating extra work on your saw. Have some one help you and as the material feeds off the back of the blade add a wedge to the kerf.
If you want to cut a 2 × 6 into 2 × 1 × 6 Go to a timber yard and buy half the quantity you need of 1 × 6 as you will not get 2 × 1 × 6 from a single piece of 2 × 6.
You will get one at 1 " and another less than 1"
If that suits your purposes do a Keifer or L/W solution.
Otherwise save your 2 × 6 for a different project an lash out for the total stock required.
Regarding the comment about the kerf closing and having to remove the splitter, just put it back on for the last half of the cut. I’ve never had a board close when I was cutting only half way through, but I have had it close on the last run.
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