It depends on whether your educational institution will be a designated one or not which you won't be able to know until CIC publish a list for designated schools in the coming months.
If your educational institution is a designated one:
No problem then, you will be able to work off-campus without a work permit starting June 1.
If your educational institution is a non-designated one:
1- Since the new regulations indicate that "Study permits will only be issued to successful applicants who are pursuing studies at an educational institution that has been designated to receive international students" In this case, Study permit holders who are studying at a non-designated institution when the new regulations come into effect will be permitted to complete their program of study, up to a maximum of three years after the regulations take effect.
2- The new regulations indicate that a student won't be able to wok off-campus unless the educational institution at which he is enrolled is a designated one, so in case your institution is a non-designated one, you won't be able to work off-campus with only your study permit, which means, you will need an off-campus work permit.
According to CIC: International students who are studying at a non-designated institution and hold either an Off-Campus Work Permit or a Co-Op Work Permit will be permitted to continue to use, and renew if necessary, those work permits until they complete their program of study, up to a maximum of three years after the regulations take effect.
So, I would advice you to keep an eye on the list of designated institution yet to come, if your institution comes as a non-designated one, you would have to apply before June 1 for off-campus work permit which will allow you to work even when the new regulations take effect.
