S Pass have not approved yet, It has been 1 month

Document submission date to HR : 30/05/2022
HR applied date: 02/06/2022

Received updated that by 22/06/2022 S Pass will be approved and signed my offer letter on 15/06/2022.

Received call on 28/06/2022 from HR stating that we are still following up with MOM. No issues just send photocopy of your passport with clear face picture. HR said by mid of July will approve and will follow up with MOM to expedite the process.

Now, should I call them again next week to follow up because my notice period with my current company is 2 months and I should join Singapore by 22/08/2022 according to the company.
Hariharan:
To apply for a work pass, there must be a valid employment contract between employer and job seeker. So I do not understand what the "offer letter" is for, which you signed during the S-Pass processing?
Regarding processing time, you are at MoM's mercy. The processing time varies between two weeks and two months in most cases (special circumstances might deviate from this). Unclear or missing documents (as in case of your passport) will of course delay things further.
You can call your employer again and again, but this will not change anything at MoM. A better course of action is to be patient and wait, resign from your previous job only after your work pass is approved - and have a Plan B in case the application is rejected altogether!