しっくり by itself is an adverb meaning something like "nicely", "perfectly", describing how something fits well with something. But it's almost always used in combination with 来る, anyway.
～(と)来る after certain adverbs is something like "comes off to me as ～", "strikes me as ～", "It occurs to me that ～", etc. It's the fifth definition of デジタル大辞泉:
So しっくりと来ない literally means "it does not strike me as perfect" or something similar. It's a set phrase, and possible translations would be "I feel something is wrong", "I somehow feel uneasy", "It's kinda out of place", "It seems a bit off", etc. As mentioned in the comment, the cause of such a feeling is not mentioned in your excerpt.
～て早々【そうそう】 means "No later than ～", "As soon as ～".