This post addresses the remaining mentions of thanks and thanksgiving in Paul's epistles. In 2 Corinthians 1, Paul comes close to one of those introductory thanksgivings we saw before. Paul essentially indirectly gives thanks for their salvation, attributing it solely to grace.
In 2 Corinthians 4, there is related mention of thanksgiving. Here, however, we note that Paul connects us giving thanks and God being glorified. Furthermore, the thanksgiving is motivated by the grace of God - his unmerited favor of us.
In the final passage, in 1 Thessalonians 3, Paul asks how enough thanks can be given to God for the Thessalonians - and that is even despite their imperfect faith! All of these relate to the same theme as the introductory passage: Paul's thanksgiving to God for the salvation of those to whom he preached.
2 Corinthians 1:8-12
For we would not, brethren, have you ignorant of our trouble which came to us in Asia, that we were pressed out of measure, above strength, insomuch that we despaired even of life: but we had the sentence of death in ourselves, that we should not trust in ourselves, but in God which raiseth the dead: who delivered us from so great a death, and doth deliver: in whom we trust that he will yet deliver us; ye also helping together by prayer for us, that for the gift bestowed upon us by the means of many persons thanks may be given by many on our behalf. For our rejoicing is this, the testimony of our conscience, that in simplicity and godly sincerity, not with fleshly wisdom, but by the grace of God, we have had our conversation in the world, and more abundantly to you-ward.
2 Corinthians 4:13-16
We having the same spirit of faith, according as it is written, I believed, and therefore have I spoken; we also believe, and therefore speak; knowing that he which raised up the Lord Jesus shall raise up us also by Jesus, and shall present us with you. For all things are for your sakes, that the abundant grace might through the thanksgiving of many redound to the glory of God. For which cause we faint not; but though our outward man perish, yet the inward man is renewed day by day.
1 Thessalonians 3:6-10
But now when Timotheus came from you unto us, and brought us good tidings of your faith and charity, and that ye have good remembrance of us always, desiring greatly to see us, as we also to see you: therefore, brethren, we were comforted over you in all our affliction and distress by your faith: for now we live, if ye stand fast in the Lord. For what thanks can we render to God again for you, for all the joy wherewith we joy for your sakes before our God; night and day praying exceedingly that we might see your face, and might perfect that which is lacking in your faith?