Your sentences need to be rephrased.
'Your stint starts in May and ends in September.'
'Your stint is from May to September inclusive.'
Even then, they are not precise enough.
You need to stipulate the dates more clearly:
'Your stint is from 1st May to 30th September.'
'Your stint is the months of May to September inclusive.'
Inclusive means including both of the months stated.
It is usually used for dates rather than months:
'I am available from 1st May to 30th September inclusive.' This means
I am available on 1st May and 30th September and all the days in between.
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