If you go to Hong Kong you can get a new Chinese visa. For many EU citizens the cost is HK$150 for a single entry (pay more for more entries and faster service), while for a US passport holder the cost is HK$1100 (same price for any visa; pay more for faster service). See http://www.fmcoprc.gov.hk/eng/zgqz/blsjfy/ Currently you might be limited to a double entry China visa in Hong Kong, although in the past multi entries have been available.
Applying for a new visa in Hanoi would be difficult, while HCMC it is reportedly easier.
If you have a double entry, it would be wise to go to Vietnam first; then go back into China and make your way to Hong Kong. Then in Hong Kong get a new Chinese visa.
There is also the China Visa Extension Thread which gives good advice about where to extend, and where not to.