I would say that the following verses do support that the church is not only the local church but the body of Christ.
1 Corinthians 15:9
For I am the least of the apostles, unworthy to be called an apostle, because I persecuted the church of God.
Ephesians 1:22-23
22 And he put all things under his feet and gave him as head over all things to the church, which is his body, the fullness of him who fills all in all.
Ephesians 5:23-27
For the husband is the head of the wife even as Christ is the head of the church, his body, and is himself its Savior. Now as the church submits to Christ, so also wives should submit in everything to their husbands.
Husbands, love your wives, as Christ loved the church and gave himself up for her, 26 that he might sanctify her, having cleansed her by the washing of water with the word, 27 so that he might present the church to himself in splendor, without spot or wrinkle or any such thing, that she might be holy and without blemish.
Ephesians 5:29-30
For no one ever hated his own flesh, but nourishes and cherishes it, just as Christ does the church, 30 because we are members of his body.
Colossians 1:18
And he is the head of the body, the church. He is the beginning, the firstborn from the dead, that in everything he might be preeminent.
Colossians 1:24
Now I rejoice in my sufferings for your sake, and in my flesh I am filling up what is lacking in Christ's afflictions for the sake of his body, that is, the church,