Look at the relevant provisions of the social insurance law.
In addition, you started your job in May and signed the labor contract in September, so in these three months, your double salary should be more than two months.
The employer who has not signed a labor contract shall pay you double salary, social security and economic compensation for the termination of labor relations from the second month (for legal provisions, see Articles 82, 38 and 46 of the Labor Contract Law and Articles 6 and 27 of the Regulations for the Implementation of the Labor Contract Law). The focus is on evidence, which needs to prove that you have a labor relationship with the employer. As for the number of hours worked, the principle of "inverted burden of proof" can be adopted, and the employer will issue it in arbitration or litigation, because everyone has to fill in the employment form when working in a place. This also proves your working hours, you have to sign the salary, and the salary payment form has to be issued by the employer to prove your monthly income.
"Inverted burden of proof" widely exists in the field of labor law. Articles 6 and 39 of the Labor Dispute Mediation and Arbitration Law, Article 13 of the Supreme People's Court's Interpretation on Several Issues Concerning the Application of Laws in the Trial of Labor Dispute Cases, Article 9 of the Supreme People's Court's Interpretation on Several Issues Concerning the Application of Laws in the Trial of Labor Dispute Cases, Article 6 of the Supreme People's Court's Provisions on Evidence in Civil Litigation, Article 2 of the Notice of the Ministry of Labor and Social Security on Establishing Labor Relations, wage payment vouchers, social security. The employer may be required to bear the burden of proof.
If you want to fully protect your legitimate rights and interests, I suggest you look at the Labor Law, the Labor Contract Law, the Regulations on the Implementation of the Labor Contract Law, and the Social Insurance Law, so that you can know in what ways the employer has violated your rights and interests, which will benefit you for life.
If the employer infringes on your legitimate rights and interests, pay attention to collecting evidence. This is very important if there is arbitration or litigation in the future.
See Article 9 of the Interim Provisions on Wage Payment and Article 50 of the Labor Contract Law for the time of wage payment when the labor contract (or labor relationship) is dissolved. If the payment is not made on time, it can be handled according to Article 3 of the Measures for Economic Compensation for Violation and Termination of Labor Contract 10 or Article 85 of the Labor Contract Law.
If you carefully look at the content of my answer above, you can fully understand the meaning of my answer.
My Baidu space has the above legal provisions, you can check it.