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The prime challenge in unsupervised domain adaptation (DA) is to mitigate the domain shift between the source and target domains. Prior DA works show that pretext tasks could be used to mitigate this domain shift by learning domain invariant representations. However, in practice, we find that most existing pretext tasks are ineffective against other established techniques. Thus, we theoretically analyze how and when a subsidiary pretext task could be leveraged to assist the goal task of a given DA problem and develop objective subsidiary task suitability criteria. Based on this criteria, we devise a novel process of sticker intervention and cast sticker classification as a supervised subsidiary DA problem concurrent to the goal task unsupervised DA. Our approach not only improves goal task adaptation performance, but also facilitates privacy-oriented source-free DA i.e. without concurrent source-target access. Experiments on the standard Office-31, Office-Home, DomainNet, and VisDA benchmarks demonstrate our superiority for both single-source and multi-source source-free DA. Our approach also complements existing non-source-free works, achieving leading performance.