H1 transfer with Multiple employers

[chg123]

I got stamped on H1 visa on January 2017 and travelled to US on March 2017. Based on new employer A's offer letter and promise that they will file new petition very soon, I have resigned from current organization who is petitioner for my H1 visa and last day in current organization in US is completed 1 week back(i.e on 9th May 2017) but another payslip may generate on 18th May 2017. My current H1 visa is valid till 31st July 2018. Due to few issues, my new employer A delayed in filing new H1 petition for me and they are telling me that they will file new H1 petition in few days and I am waiting for the same. Once new employer A filed new H1 petition and then once I started working for new employer(employer A) in a week, can I apply another new H1 petition through employer B or employer C ? Should I wait till new employer A petition should be successfully approved from USCIS? Considering current H1 visa is valid till 31st July 2018 and 60 days grace period from last day of previous organization, can I apply new petition for employer B or employer C without employer A petition approved? Please suggest. If yes, frequent change of employers within few weeks on H1B status, it will affect on any future H1 extension approvals or I-140 approval?