Go to any travel agent in Hong Kong and they can get you a 3 month multi entry visa. No proof is required, nor is an invitation letter required either. Visit ChungKing Mansion in Kowloon, Kowloon is in my opinion the go to place in HK. It is on Nathan road, Tsim Sah Tsui East line, not hard at all to find. Only problem with this visa is you must leave China every 30 days.

Anyhow, once there the building is loaded with visa agents, travel agents, as well as guest houses of various sorts, most super cheap.

You will find all you need there. Once you have that F visa of 3 months duration, I can not say whether or not it can be extended, I doubt it, but posibly. However travel agents in HK can also procure you a longer term F visa, for a higher price. Cost of a 3 month F is not too bad, between 1,300 to 1,900 yuan. at Chunk King Mansion you'll pay around 1,600.

Just so you know, the freeze on L to F visas is temporary, meaning that it will expire after the Chinese new year. If I were you, I would get my L visa extended for one month, which you can do twice, the cost is only 130 yuan in most places for a one month L extension, for zero entries. This can be done at any entry and exit burea in whatever city you are in. All you need, hotel temporary residence card, which by law and hotel you stay at must give you, passport, a passport size photo, and a photocopy of your old visa and passport page. Piece of cake really. Some places will ask for a bank slip saying how much money you have in the bank, to show you can afford to live here. However, if you tell them you simply get sent money via western union every one or two weeks, and pull out a big wad of cash, well, cash is proof enough, hard to argue with that as proof you can afford staying here.

Call or email any visa agent in mainland China, they will tell you the same, that after Chiense new year ban on L to F will end.

My advice, get your current L visa extended, than in a month, use a mainland visa agent to get an F for 6 months, cost around 3,500. Or you can buy yourself a cheap invitation letter, common practice, and apply in HK for yourself, might save you some money, costs around 1,400 for next day service I believe if you apply yourself.

Hope I have been helpful. If you want the names of some trustworthy mainland visa agents, or ones in HK, send me an email. Take care, and good luck.