Did Paul preach a different kingdom that was contrast to the kingdom Jesus preached? I do not believe the kingdom Jesus preached stands in contrast to Paul's gospel. However, it is interesting to notice there are differences to explore, but both are the same theological kingdom reality.
Jesus taught the kingdom was the fulfilling of the OT promises and saving rule of God (as mentioned in our Lecture #2). Jesus taught the kingdom was fulfilled and ruled in him, yet with an already-not-yet explanation (i.e. inaugurated but not yet consummated). Jesus heralded the coming salvific work of God and preached the kingdom was present wherever he was, and yet was to be climatically realized in the future (see Luke 4:43, 6:20, 7:28, 8:1, 10; 9:2, 11, 27, 60, 62; 10:9, 11; 11:2, 17, 18, 20; 12:31-32; 13:18, 28, 29; 14:15; 16:16; 17:20-21, 18:16-17, 24-25, 29; 21:10, 31; 22:16, 18; 22:29-30; 23:42, 51)
Paul taught the kingdom was the fulfilling of OT promises and saving rule of God, but in salvation and judgment in Christ. Since Jesus came and proclaimed the kingdom was fulfilled and supremely rules in him, Paul taught Gentiles the kingdom of God is of righteousness, peace, and joy in the Spirit (Rom. 14:17) Living in the Spirit is the key to inheriting the kingdom of God (those in the kingdom are saved; those not in the kingdom are judged. 1 Cor. 4:20, 6:9, 10; 15:50; Gal. 5:21; Eph. 5:5). Paul also emphasized an already-not-yet explanation for believers who will one day be in God's kingdom (1 Cor. 15:24; Col. 1:13, 4:11; 1 Thess. 2:12; 2 Thess. 1:5; 2 Tim. 4:1, 8).
Summarizing the differences instead of direct contrasts, it is clear Jesus' kingdom was preached in the present-direct fulfillment and reality of the coming of the kingdom of God. Paul's kingdom, however, was preached in the present reality of the already-came kingdom of God in Christ. Thus, the kingdom in Paul's letters are expressing the same salvific work that is fulfilled and supremely ruled in the Son, just as the Son preached.