Can I cancel then renew a PRC tourist visa?

I am an American and I am planning on moving to Taiwan and teaching English for a year (possibly more). I currently have a one year multiple entry tourist visa for the PRC. I plan to come to Taiwan some time in June but I currently possess is valid until early July. Now I know that the PRC will generally not issue multiple entry visas outside of the applicants country of citizenship (and through experience with friends in the last few months sometimes not even double entry visas). Now this isn’t that big a problem for people not from the US but for Americans all PRC visas cost the same high price (~US$140) no matter the length of validity or number of entries. Since I fully intend to travel to China at least two or three times while I am in Taiwan I would really like to get a multiple entry visa which means applying in the United States. This is why I am wondering if I could have the PRC consulate cancel my visa and reissue another, either in one step or two. I realize they might not do that which is why I ask but the alternatives kind of suck (paying US$140 for each visit or mailing my passport to the US to have someone apply for me).

Note: I don’t think it matters for this but I am not of Chinese or Taiwanese decent.

We usually make short visa runs to Hong Kong, sometimes you can comeback the same day and you can also get your China visa there. I don’t have a lot of experience but I think everybody else is sleeping right now.

I heard nowadays you can use your visa free entry and then get a resident card when you get a job (ARC, search that in the forum).

And as long as I know you can’t teach on a visitor visa, single or multiple, you need a resident visa (and card) with a work permit but you get those here, so you may be wasting money on that.

So look it up in the forum, I’m sure you’ll get a lot of answer when later today.

Also just to clarify we have the same mindset.
PRC = China
ROC = Taiwan

Yes, I am talking about getting a visa to the PRC, mainland China. I am not worried about getting into ROC Taiwan.

Also I am aware I could get a PRC visa in Hong Kong, my only concern is getting a multiple entry visa which would mean getting it before I leave the US.

No, you cannot. It has to expire on its own before you can get anything new in the passport. Troublesome.