There are a number of issues around the plan you propose.
1. Any offering must have a maximum amount or a maximum number of shares. This is basic disclosure for potential investors. A well-structured offering with common investor protections also has a minimum amount to be raised before you cash the checks.
3. Be careful about losing the exemptions under which you are doing the offering. If you are selling to investors in more than one state you're subject to the Federal regulations (many of the exemptions are limited to $1 million per year). If you are selling only in one state make sure you comply with your local rules.
4. The biggest reason not to try this is most investors wont play. You need to have made some pretty serious progress to justify a big step up in valuation in close proximity to the last investment. I usually counsel at least six months between the close of a round and increasing the valuation because of investor resistance.
5. Another reason for not doing it is that it gives the impression you're more focused on financial gamesmanship than running and building the company. As CEO you should focus on creating underlying shareholder value, not extracting marginal dollars from your last investor. If you're about to make big strides shut down the old offering, wait at least several months and come back with a new offering based on the meaningful change in the business. If you're doing common now and want a big increase in value, make the next round preferred (small increases in value close in time rarely succeed). Otherwise, keep it simple, use your achievements as a way to bring in capital quickly (you can tell people you're closing the round in X days or weeks because you believe the value has increased but make sure you really do it or you'll lose credibility) and go build a great company.
Good luck with the raise.