Colonial smiths used lye solution to neutralize the acid. I've tried it and it adds a redder tone to the wood to my eyes. A table spoon to a bucket of water is enough. Wear gloves! It's not a strong enough solution to cause harm, but you won't be able to grasp anything!
You are correct on the redder tone. That is a function of the strength of the caustic being used.
I had done some work on that in the lab comparing the different caustics and the colors produced.
The color of iron oxide can vary over a wide range of the "earth colors". Molecules of iron oxide will hold molecules of water. The amount of water attached to each molecule of iron oxide will determine its color. If very strong sodium, or potassium, hydroxide is used the red gets so intense that in the form of a wet paste it appears black.
If you swab the nitrate of iron stain on a stock and stand the stock in a damp coroner of the basement you get mainly yellow and some orange and brown. Then heat it gently to drive off some of the water attached to the iron oxide and it will go into the brown colors. But to use enough heat to drive off all of the water to get red colors you end up scorching the wood to a brown color.