According to Manu Smriti -- NO.
See the following verse for e.g.:
10.4. Brahmana, the Kshatriya, and the Vaisya castes (varna) are the twice-born ones, but the fourth, the Sudra, has one birth only;
there is no fifth (caste).
If a Sudra can attain higher caste in the current life then it would not have said for him that he has only one birth. Because he will then be eligble for performing necessary sacrifices like Vratyastoma and get his Vedic sacraments, which will be his second birth.
And, also see:
10.122. But let a (Sudra) serve Brahmanas, either for the sake of heaven, or with a view to both (this life and the next); for he who is
called the servant of a Brahmana thereby gains all his ends.
10.123. The service of Brahmanas alone is declared (to be) an excellent occupation for a Sudra; for whatever else besides this he
may perform will bear him no fruit.
So, again, if the text agrees with a Sudra getting a higher caste in the current birth, then it would not have said 10.123, because then for a Sudra serving the Brahmins would not have been the only fruitful occupation.
and, finally see this:
10.126. A Sudra cannot commit an offence, causing loss of caste (pataka), and he is not worthy to receive the sacraments; he has no
right to (fulfil) the sacred law (of the Aryans, yet) there is no
prohibition against (his fulfilling certain portions of) the law
This again is contradictory with the statement that "a Sudra can attain higher caste in the current life" because then it would be verily possible for him to loose his caste.
So, for the given verse 9.335 - translation wise, the Wisdom Library's one is precise, because the verse does not explicitly mention which birth but overall interpretation wise, the other translation/interpretation is correct, because the verse is certainly not talking about attaining higher castes in the present life itself.
Because, the text, Manu Smriti, does not support such an idea at all.