δεξιῇ ἠσπάζοντο ἔπεσσί τε μειλιχίοισι
From the three uses of the verb ἀσπάζομαι in the Odyssey, we can see that it is used in contexts of welcoming someone who has just arrived after a journey. Nestor and the Pylians greet Telemakhos upon his arrival in Pylos (Odyssey 3.39), Autolykos welcomes his grandson Odysseus after he arrives for a visit (Odyssey 19.415: the whole line is very similar, χερσίν τ’ ἠσπάζοντο ἔπεσσί τε μειλιχίοισι), and the slave women welcome Odysseus home after Eurykleia has told them of his arrival (Odyssey 22.498). Thus its use here is one more indication that spying missions and ambushes share the traditional thematic language also used for journeys in these epics, and we can understand such “Odyssean” language as related to the theme, rather than any indication of “lateness.”
See above on 10.288 for more on the meaning of μειλίχιος. On that line, it is used to describe a muthos, and here it is used with epos. Although we may translate both words as ‘word’, Richard Martin has demonstrated that there are important distinctions between the terms in Homeric diction: epos is the unmarked term for “utterance,” while muthos is the marked term for a public speech-act, a “performance” of sorts that conveys authority and power (1989:22–30). When we compare the different contexts of the adjective μειλίχιος in these lines, then, we can observe that Tydeus’ muthos was indeed a public, performative speech in which he was at least trying to exert his authority, whereas these words of welcome carry none of those connotations. Martin also notes that “a muthos focuses on what the speaker says and how he or she says it, but epos consistently applies to what the addressee hears” (1989:16). Thus words of welcome are appropriately epea, since they are particularly heard as such by the addressee.