At the Reformed Baptist church I attended the preacher would say if anyone had never come to Christ they should see him after the service.
The bible Presbyterian Church I now attend will on some Sundays (like yours usually Communion Sundays and on holidays ) say something very similar..to see him after the service or call for an appointment to meet with him.
I believe this is a sound way to offer the gospel as it elimates a non sincere emotional response.
I think it's absolutely appropriate to administer Communion to new Believers. The Sacrament is a Means of Grace (even if we lay aside the Presbyterian doctrine of "Spiritual Presence", and we take a strictly Baptistic view that Communion is observance of a Symbolic Ordinance -- it is still one of God's means of imparting Grace to a believers spirit for a believer to obey the Lord's ordinances!!).
Provided that they examine themselves to confess known Sins, and are submitted to their Elders -- Why should we deny participation in the receipt of sacramental, sanctifying Grace to those who (arguably) need it most, New Believers?
Of course, I should admit in the spirit of Full Disclosure that I'm nearly a Paedo-Communionist myself -- I don't believe in Communing the Infants as some paedocommunionists do, but I don't believe in withholding the Sacrament until the adolescent years, either. In the absence of any Biblical case law on the matter of which I am aware, I advocate the continuance of the Hebrew custom for the Passover Seder -- administering the Supper to covenant Children as soon as they are old enough to explain the meaning of the sacrament to their Elders.
JMHO, as always.
best op