I do think Paul is consistent with the teaching of Jesus, in that it is sin to initiate the destruction of a marriage (whether by adultery or other sexual sin, or abandonment), but that a person who has not been responsible for their marriage breakup is exonerated from the guilt of such.

Hence, I concur with Al Maxey's view on the 'exception clause' in Matthew 5 and 19. Paul, in 1 Corinthians 7:15 is clearly exonerating a believer from the guilt of a sundered marriage, and there would be no guilt in a following remarriage if reconciliation was unlikely. The highest aim is reconciliation, as Paul states in 1 Corinthians 7:10-11.
I believe that Jesus only states hyperbolically that a woman who has been divorced from her husband wrongly 'commits adultery' with a new spouse because that new spouse in a sense is condoning the hardheartedness of the original spouse, hindering reconciliation.

Of course, for a believer who is abandoned, there is very little hope of reconciliation, and Paul cannot command such from the unbeliever, hence he uses the principle Jesus teaches to exonerate the believer.