I have a double entry visa but need to enter China three times. What can I do?

I have a Chinese visa valid till 20 Dec 2013 (6 months). It is a F visa, which means a business visa. It also is a double entry visa, meaning that I can only enter China twice in the whole 6 months period.
The problem is that I need to enter China three times. I arrived in China on July 10th, I will leave China on July 31st to go to Europe and come back to China on August 17th (here I will already have consumed the second entry). And I need to go from China to Hong-Kong on Augst 24th and back to China on August 30th. Of course this last entry will not be possible with my current visa. What can I do, then?
The only option I can foresee is that on July 31st when I travel to my country, apply for a new Chinese visa (a new double entry visa). This would solve the problem. However, can I apply for a new visa if my visa is still valid and still has one entry left?
If the answer was yes, then it is solved I apply and that is it.
If the answer is no, then, if I had consumed the two entries before the new application, could I apply for a new one? This way, although the visa was still valid in terms of dates, it would not have entries left. If yes, then I can consume the second entry before travelling to Europe and then apply for the new visa.
But if not, what can I do? Can I cancel the visa and apply for a new one? Is there such a thing as cancelation? Or What option is left for me?
One last issue: once I find the way to apply for a new visa in these circumstances, should the new visa be Business or Tourism? I plan to combina both in my second period in China. If for business, could I use the same original letter from the company inviting me to come to China for two months, that I used for the first visa I got? Should I get a new letter from the company? Better to apply for a tourist visa? But if I do that, they could see that in the letter for the original visa the invitation for business was for two months, and now I am applying for a tourist visa for a period that correspond to the one I was supposed to be doing business. Or this really does not matter?
The case is very complex, I feel lost, and action need to be taken immediately as the dates are approaching.
Best Answer
I have also successfully applied for a new visa while the old one still there in Hong Kong. In Hong Kong is very easy to obtain Chinese visa. Just go to a travel agency, I paid around 150 HK dollars and I get a new visa in two days.
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Answer 2
I have successfully applied for a new tourist visa while holding an old one. As it turned out the old visa had a few minutes left on it when we arrived, the passport official simply cancelled it at that point (entries didn't matter as these were multi-entry visas.)
While I have never had either a double-entry or a F visa I would expect things to work the same.
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