Social security found another company this month. Can another company reissue it next month?

I can't. You can complain to the social security department and ask the unit to pay the social security at that time, but it is impossible to pay the social security across companies. Some cities can't pay it next month according to the social security policy. As for the evidence, as long as you can prove that you are an employee of this company, such as labor contracts-salary slips, attendance records.