You were quite free to research this base period thing and decide when to file your claim, as in wait a week or two till the quarter changed. In many states, the quarter is not changed on the actual first day of the month, but after the first full week of the first month.
Of course you'd have been a week or two later in getting your first unemployment checks, with no reimbursement for those weeks. In my state, usually the unemployment workers are happy to tell people about the quarter system and how it operates. You can certainly decide to wait a week or two to get a bigger claim in place. But if you filed on line, of course, you didn't have this resource. But you were still free to research the unemployment system in your state and make a decision about what quarters you'd be set up from, and whether you should wait.
The "alternative base period" dealie is usually called into play when the person would not have a claim at all without using the most recent quarters. But even if it was a viable possiblity in your state, when you were given your monetary determination, you'd have had to file an appeal of this as soon as you got it, requesting that your state use the alternative base period because it would pay you more money. If you didn't do that within 15 days, it won't happen.
But since you chose to file the claim when you did, believe you me, it is NOT worth the effort to try to ask for this claim to be cancelled and a new claim to be taken. You'd have to pay back this whole claim, present compelling evidence why you were not...oh well. Let's just say that while it might be done under some circumstances....it probably won't happen in this case even if you go back and request it.
So, as Patti said, next year those wages will be there. This claim, once filed will be in effect for a year from the date of filing. If you do not have intervening work, you'd qualify for any extensions at this pay rate, no new claim at the end of the year. SO the best thing you can do is find another job quickly and not need this lower weekly amount unemployment this year.