rainy spells just before the rainy season
Refers to the intermittent rain that falls in the period leading up to the official start of the rainy season (tsuyu) in Japan, typically in late May to early June.
This year the pre-rainy-season rains came early, and we've already had a string of rainy days.
Because of the pre-rainy-season rain, the laundry won't dry.
走り here means 'beginning' or 'first signs', not 'running'. It is used in other seasonal expressions like 秋の走り (early signs of autumn).
Standard kanji spelling for this seasonal expression.
Kana-mixed spelling; the kanji 走り is replaced by hiragana, which is also seen in practice.
Compound of 梅雨 (tsuyu, 'rainy season') + の (possessive particle) + 走り (hashiri, 'first signs, beginning'). The word 走り derives from the verb 走る (hashiru, 'to run'), metaphorically extended to mean the first appearance or onset of something.