A Woman Before Eve?

I've always been told and raised to believe that Eve was the first woman created by God. This may still be true. But I can't help but bring up the possibility based on the some of the verses of there being someone before Eve. I'm not saying this is the case, I'm only asking what you think.

These verses come from Genesis (KJV).

For your reference:
Genesis Chapter 1
Genesis Chapter 2

1:26 And God said, Let us make man in our image, after our likeness: and let them have dominion over the fish of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creepeth upon the earth.

1:27 So God created man in his own image, in the image of God created he him; male and female created he them.

Okay. Here in chapter 1, God has created humans (both male and female are created on day 6). Boom. There they are. Both existing.

Now lets go further into Chapter 2 where Eve is created.

2:21 And the Lord God caused a deep sleep to fall upon Adam, and he slept: and he took one of his ribs, and closed up the flesh instead thereof;

2:22 And the rib, which the Lord God had taken from man, made he a woman, and brought her unto the man.
2:23 And Adam said, This is now bone of my bones, and flesh of my flesh: she shall be called Woman, because she was taken out of Man.

So if Eve is created at the end of Chapter 2, who was the first woman created in Chapter 1?

But it also sounds like on day 6, a man and a woman were both created in the same day. Maybe Adam named all of the animals in that same day when he was by himself and it only took the same day to realize he was lonely and needed a companion so God gave him one (Eve) but that's a lot for one day. But on day 6 male and female were created, that's scripture.

So what do you think? Was there a woman before Eve created at the time of Adam on day 6? Was Adam created, explored Eden and named all of the animals, etc... fell asleep, had a rib removed and Eve created all in a single day (day 6)?

It was just interesting to me that man and woman were created in the same day from what sounded like the same time. Then in the next chapter we hear that Eve was created to keep Adam company. There's a lot of things going on in a single day.

My thoughts?
I think it's probable that the Bible is going back and discussing in detail how it actually happened. Maybe Chapter 1 was a summary as a whole and Chapter 2 is more of a detailed look as to how it actually took place.

What do you think about this topic?