We would need something called a scripture to believe this. Since there are none, then we don't believe anything like that.
The heart of man is refered to the inner-man or spirit of a man. Jesus said you must be born of spirit and all things become new.
Things don't become new over time and there is no greater "Savlation" over time unless we take the definition of Salvation and say we can be physically healed over time.
So we just don't slowly recieve a new spirit, or slowly get saved (As in eternal life)
We can be transformed by the renewing of our mind though (soul) and we can recieve with meekness the engrafted Word which is able to save our souls (Mind) We can get more in tune with our born again spirit and the Holy Spirit, not being as flesh ruled but (spirit) Spirit ruled.
\Salvation is a process because it is spoken of in Scripture in three tenses, past present an future, have been save, are being saved, and shall be saved. That is an ongoing process.
NKJ Ephesians 2:8 For by grace you have
been saved through faith, and that not of yourselves; it is the gift of God, (Eph 2:8 NKJ)
NKJ 1 Corinthians 1:18 For the message of the cross is foolishness to those who are perishing, but to us who
are being saved it is the power of God. (1Co 1:18 NKJ)
NKJ Matthew 24:13 "But he who endures to the end
shall be saved. (Mat 24:13 NKJ)
Paul's statement, "us who are being saved" shows that it is a process.
Since the new heart is a metaphor for the inner man and the inner man isn't instantly changed when one believes it would seem that it is a process. People who become Christians don't instantly become mature in the faith that is something that takes time. We can see this in Paul's letter to the Corinthians.
And I, brethren, could not speak unto you as unto spiritual, but as unto carnal, even as unto babes in Christ. 2 I have fed you with milk, and not with meat: for hitherto ye were not able to bear it, neither yet now are ye able. 3 For ye are yet carnal: for whereas there is among you envying, and strife, and divisions, are ye not carnal, and walk as men? (1Co 3:1 KJV)