The expression διὰ παντός means
always, continually, constantly (BDAG, "διὰ", A.2.a)
This is a formulaic adverbial phrase, but it isn't really so hard to arrive at from the literal meaning of the two words if one understands διὰ as "throughout, through, during" (A.2.a). The following word is the genitive form of the adjective "all" (the genitive being required by the preposition). BDAG additionally points out that the expression likely abbreviates a longer phrase: "sc. χρόνου", i.e. "one may supply 'time'". We then have "throughout all [time]" = continually.
In addition to Hebrews 13:15, see Matt 18:10; Mark 5:5; Luke 24:53; Acts 2:25, 10:2, 24:16; Rom 11:10; Eph 4:16, 6:18; 2 Thess 3:16; Heb 2:15, 9:6.
The OP proposed an alternative interpretation, "through all (things)". This is conceivable, taking παντὸς as a substantive, but unlikely in this case in light of the very common temporal use διὰ παντός described/exemplified above. Furthermore, παντὸς is singular. It seems the plural form -- διὰ πάντων -- is indeed able to carry the sense indicated by the OP. Unfortunately the only NT example is contained in the rather cryptic characterization of God the Father in Eph 4:6,
ὁ ἐπὶ πάντων καὶ διὰ πάντων καὶ ἐν πᾶσιν.
who is over all and through all and in all. (ESV)
The LXX offers a few more examples, but the phrase is apparently most at home in descriptions of the transcendent. In Wisdom 4:6, σοφία ("wisdom") is described as the one who,
διήκει δὲ καὶ χωρεῖ διὰ πάντων διὰ τὴν καθαρότητα
pervades and penetrates all things because of her pureness. (NETS)
@Jacob -- an interesting idea. I added an addendum to the answer to address this.