Yes exclusively. The word also doesnt change that fact, it just means as some elect were already converted and became believers, there are more to be converted and become believrs, especially among the Gentiles. Paul was sent to the Gentiles
2 Tim 1:11
Whereunto I am appointed a preacher, and an apostle, and a teacher of the Gentiles.
2 Tim 4:17
Notwithstanding the Lord stood with me, and strengthened me; that by me the preaching might be fully known, and that all the Gentiles might hear: and I was delivered out of the mouth of the lion.
Therefore I endure all things for the sake of the elect, that they also may obtain the salvation which is in Christ Jesus with eternal glory.
Salvation can’t be exclusive to the elect if Paul desires for them to ALSO obtain salvation. By saying ALSO the Holy Spirit is indicating that others beside the elect have obtained salvation.
You will have to rethink your claim of salvation being exclusive of the elect since we plainly see that Paul had to endure suffering in order for the elect to ALSO obtain salvation.
Question:
Why would Paul have to endure suffering for the elect to ALSO obtain salvation if they were already saved?