The main issue here to answer this question is that we don't know anything about Óðr. He is only mentioned in :
- the paragraph you quoted from the Gylfaginning;
- stanza 25 of the Völuspá:
Þá gengu regin öll á rökstóla,
ginnheilug goð, ok um þat gættusk:
hverr hefði lopt allt lævi blandit
eða ætt jötuns Óðs mey gefna.
which translates to:
Then sought the gods | their assembly-seats,
The holy ones, | and council held,
To find who with venom | the air had filled,
Or had given Oth's bride | to the giants' brood.
- In Snorri's Skáldskaparmál:
Ok hér hefir Einarr enn kennt svá Freyju at kalla hana móður Hnossar eða konu Óðs, svá sem hér:
- Eigi þverr fyrir augna
Óðs beðvinu Róða
ræfrs, eignisk svá, regni
ramsvell, konungr elli.
which translates to:
And here Einarr has further periphrased Freyja so as to call her Mother of Hnoss, or Wife of Ódr, as standeth below:
The shield, tempest's strong roof-ice,
With tear-gold is unminished,
Eye-rain of Ódr's Bed-Mate:
His age the King so useth.
the latter part being taken from a skaldic poem: Einarr Skúlason's Øxarflokkr.
He also appears in a small amount of other skaldic poems but every time, as in the few examples above, as a kenning for Freyja (i. e. Freyja being referred to as "Óðr's wife").
So, basically, he is only known as married to Freyja and as being on a "long trip", so there seem to be no original source explaining which trip he is undertaking.
As you mentioned there has been a lot of speculation on Óðr being in fact Óðin, and Freyja being Frigg but as Hollander (1950) mentions in his article The Old Norse God Óðr, the fact that Freyja is weeping for him and go on a search for him, doesn't really fit the association as Frigg knows all fate, including that Óðin dies during Ragnarok, (so why would she weep for him?) and that Hnoss would not be their only child.
As a side point, and to make things more complicated, the adjective óðr can be found frequently in skaldic poetry as it simply means furious.