But keep in mind that you won't have a mortgage payment the following month as mortgages are paid in arrears. example: you close September 15 and you pay at the closing interest & amortization from the 15th thru end of September. You won't have another payment due until November 1 and that covers your ownership from the the beginning to the end of October.
Generally speaking, if you close at the beginning of the month you will bring more money to closing. The lender will require you to pay (at closing) the interest on the loan from the day you close until the last day of the month you close. For example, if you close on September 15th - as part of your closing costs you will need to pay interest from September 15th through September 30th. That's why most buyers prefer to close closer to the end of the month. It minimizes the amount of money they have to bring to the closing table.
Generally speaking you need to bring less money to the closing table if you close at the end of the month. You need to pay interest per day for every day left in the month of your closing. So if it is the 29th you only pay for the 30th and/or 31st, as opposed to a close on the 5th when you would gave another 25 or so days of interest left to pay. I'd try to avoid closing on the very last day though. One delay in closing will put you into the next month.