OP's GC expiration date isn't until June. If OP starts the new job before the expiration date then the new employer should NOT ask for the I-751 Receipt notice.
In the case of a new job your only options are to get a i551 stamp from USCIS office or wait until you get the receipt notice in the mail.
The primary thing is to file the i751 on time (before CGC expires) and the application is accepted and not returned. It will take weeks and weeks to learn whether the i751 was accepted and even longer to get an unsuccessful application returned to you. You will know when i751 is accepted when your check is cashed.
Main reasons i751 are not accepted:
Payment issues, some i751 have had credit card not accepted by the bank; I recommend a personal check or money order.
Filing too early (not an issue here)
Including the wrong payment amount (Check USCIS web site for current i751fee plus the biometric fee and how to make out the check) total of i751 fee plus biometric fee.
Sending i751 package to the wrong address. Check the USCIS web page for current addresses for the shipping method you are using.
Not using the current i751 version, please check the USCIS website to download the latest version of the i751 form.
Not signing the i751 form or not signing correctly, make sure the conditional green card holder signs as the petitioner and the USA spouse only signs as the spouse.