Your previous organization can help you by providing a statement on the company letterhead. Additionally, I assume you have an ID card from your previous company. You can also request an experience letter or a relieving certificate that includes the necessary details.
I would like to emphasize that an employee should not resign from their current company unless they have received an offer letter from the new company. Upon joining and submitting the required documents within a month, you should receive an appointment letter, regardless of the size of the company.
Some companies refrain from issuing payslips or offer letters due to concerns about employees leaving or potentially negotiating for a higher salary with their current company based on an offer letter from a new employer.