οἶδα γὰρ ὅτι τοῦτό μοι ἀποβήσεται εἰς σωτηρίαν διὰ τῆς ὑμῶν δεήσεως καὶ ἐπιχορηγίας τοῦ πνεύματος Ἰησοῦ Χριστοῦ (Philippians 1:19)
I would submit that Philippians 2:1 and 3:3 are helpful in understanding Philippians 1:19. In 2:1, it's possible to understand πνεύματος as referring to a "spirit" other than God's holy spirit--but its interesting to note that C.B. Williams translates 2:1 as follows: "any common share in the Spirit." Cf. Weymouth's "any common sharing of the Spirit" and Ralph Earle's Word Meanings in the NT (page 335). Additionally, Philippians 2:1 reminds me of a similar Pauline expression in 2 Corinthians 13:14: ἡ κοινωνία τοῦ ἁγίου πνεύματος μετὰ πάντων ὑμῶν.
2 Corinthians 13:14 contains an articular use of πνεύματος whereas Philippians 2:1 is anarthrous. However, the omission of the article in 2:1 does not mean that πνεύματος there lacks definiteness. Cf. the anarthrous use of πνεύματος in Hebrew 9:14 which could be translated: "how much more will the blood of Christ, who through the eternal Spirit offered himself without blemish to God" (ESV) Moreover, Philippians 3:3 seems to be a clear reference to the spirit of God(πνεύματι θεοῦ) although the construction there is similarly anarthrous.