Yes and no - a "general call" and an "effectual call". I wouldn't say a "more persuasive" call but an effectual call through which we are made new creatures in Christ (the power of the gospel).
Part of the condemnation (I believe the primary part) is that men reject the Son. I don't mean that men must hear the gospel message, but in their rejection of God (through what is revealed) they reject the Son as the Redemption of God because God is faithful to forgive those who repent and turn to Him. This was true in the Old Testament and this is true with the arrival of the New Covenant (God is immutable and there has always been One Way and One Way offered to the World). I do not believe that God's plan of redemption changed at the Cross. It was always to save those who would believe (those "among the elect").
I do not believe that God chose to create elect and non-elect people, actively saving some (causing their faith) and actively damning others (causing their disbelief). I believe in "double predestination" only in the sense that God purposed to save some (grace through faith, all a work of God) and not to save others (both for His purposes).
The difference, I think, between my view and
@Reformed 's position (and he can correct me if I've misunderstood) is that I believe everything - to include judgment - is given unto Christ through the Son's faithful obedience to the Father even unto His death on the Cross. I believe that on the Cross we see God reconciling the world to Himself (which I believe implies placing all things under Christ - even the reprobate who will not believe).
So Christ died for the world because God loves the lost. And this love is evidenced to through those who do believe and, in the negative, it is evidenced through the judgment of those who will not believe. It is all about God.