Herbert Weir Smyth, A Greek Grammar for Colleges

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[*] 2987. The inferential conjunctions τοιγάρ, τοιγαροῦν, τοιγάρτοι, τοίνυν contain τοί, the locative of the demonstrative τό, which case had the meaning of τῷ (τῶ) therefore, on this account, so lit. by that, therein. (This τῷ is chiefly Epic, and stands at the beginning of the verse. Cp. ““τό” therefore” Γ 176, S. Ph. 142.)

τοιγάρ (prepositive; Ionic and poetic) therefore, wherefore, so then, that is surely the reason why (often to announce a purpose).

τοιγαροῦν, τοιγάρτοι (both prepositive) are more emphatic than τοιγάρ. The final syllable of τοιγάρτοι is the τοί of 2984.

τοίνυν (postpositive and post-Homeric; -νυν 2927) is transitional (now then, further) or inferential (therefore, accordingly; less emphatic than τοιγάρ). τοι<*>νυν is common when a speaker refers to something present in his mind, when he continues or resumes what he has been saying, and when he passes to a new aspect of a subject. It is often found with imperatives (σκόπει τοίνυν P. Cr. 51c).