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The Labour leader has warned the terms of reference, which have yet to be published, could be too "narrow" and residents not properly consulted.
He wants an interim report on the cause of the fire by September, followed by a wider probe into building regulations, council funding and other issues.
The presumed death toll stands at 80.
Police have warned the final figure will not be known until at least the end of the year.
The leader of Kensington and Chelsea Council, Nick Paget-Brown, which was criticised for its response to the disaster on 14 June, quit on Friday.
Retired former Court of Appeal judge Sir Martin Moore-Bick was appointed to head the inquiry on Wednesday. It will sit in public and have the powers to compel witnesses and obtain evidence.
Visiting the scene of the blaze on Thursday, Sir Martin said he believed the probe and its recommendations would be "pretty well limited" to what had caused the fire and its rapid spread.
Setting out his ambition to produce a report within a year, he said he was aware that people in the area want a broader investigation but he was not sure that his inquiry would be the right setting.
The prime minister has said she expects an interim report to be made available as soon as possible and that survivors and residents should be consulted on the terms of reference.
But Mr Corbyn said he wanted a guarantee the initial findings would be published by the end of the summer.
In a letter to Mrs May, he said the inquiry would be best served by being split into two parts to ensure "timeliness would not have to be sacrificed for rigour".
While understanding how the fire started was of the "utmost importance", he said survivors and other residents had a multitude of other questions about how their safety had come to be compromised and what could have been done to prevent the disaster.
"Whilst the detail of the specific failures at Grenfell needs to be uncovered, it is also clear that the Grenfell fire has much wider implications for national policy issues," he said.
The issues that must be addressed, he said, included:
"An inquiry within narrow terms of reference risks failing to learn all the lessons of this tragedy."
The PM has said the immediate priority is to establish the facts of what happened at Grenfell Tower and to take the necessary action to prevent a similar tragedy from happening again.
But she has said the "wider lessons" must be learn from both this catastrophe, and the inspections of other buildings around the country that followed it.