Hi OP,
Do you have a confirmed start date for work in September, or are you applying for jobs now and intending to start in September?
If you have a confirmed start date, you can apply up to 31 days before your start date:
"If you are not currently working
You can apply if you’re starting or re-starting work within the next 31 days."
www.gov.uk/30-hours-free-childcare
If the start date is early September, then you might want to call HMRC, and check whether it is likely an application in early August would be processed before the 31 August deadline:
"You can apply outside of these recommended dates [15 June - 31 July] but you might not receive your code in time. You must have a valid code by the end of the month before a new term starts."
www.gov.uk/30-hours-free-childcare
"To be able to use your extended entitlement you will need to have made a successful application and received a code by the deadlines given. If you miss the deadline you cannot use the extra funded hours, but you will still be able to use the universal hours. For example, if you do not apply and receive a code by 31 August deadline, you cannot use your extra hours in September, you will need to wait until 1 January."
www.somerset.gov.uk/education-and-families/paying-for-childcare/30-hours-extended-entitlement/
Alternatively, if you are self-employed (it doesn't say in your OP), the earnings limit doesn't apply for the first 12 months, so I'm not sure if it's possible for you to start self-employment now on a part-time basis (since you don't have to earn the minimum amount, you could take on limited work from now until September around caring for your DD?)
"This earnings limit does not apply if you’re self-employed and started your business less than 12 months ago."
www.gov.uk/30-hours-free-childcare
Even something small like selling things on Etsy?
Best of luck OP.