new: recently come into being, fresh, as opposed to what is old or former
In this meaning the word describes a thing by its age rather than its kind: it has only lately come to be, so it is fresh and unspent, standing over against whatever is old or worn out. That is why it fits both a literal picture and a spiritual one with the same force.
The literal picture is fresh wine still working in the vat, or a batch of dough just mixed. The figurative uses carry the same idea of recency: the fresh self a believer has put on, the newly cut batch of dough the church is meant to be, the covenant only lately inaugurated through Jesus.
Worth noticing is what this word does not stress. Greek has a second word, kainos, for what is new in kind or unheard of before. This word points instead to what is simply recent. The two overlap in practice, and the New Testament writers often use them side by side, but the shade here is freshness in time, not novelty of type.