Can I reduce a new Soldier for inefficiency?

First you will have to wait until the Soldier has been in the unit for 90 days. See AR 600-8-19 paragraph 10-6:

A Soldier must have served in the same unit for at least 90 days prior to being reduced one grade for inefficiency.

After that, you need to make sure you have all your documentation together. If you haven’t been counseling often, start doing it now. You need to document what corrective training has been conducted as well as the outcome. Be sure to fill in the assessment blocks on these statements. You must show documented efforts to help the Soldier improve.

Here is an extract from AR 600-8-19

a. A Soldier must have served in the same unit for at least 90 days prior to being reduced for inefficiency.
b. The CDR initiating the reduction action will present documents showing the Soldier’s inefficiency to the reduction authority. This may include:
(1) Statements of counseling and documented attempts at rehabilitation by chain of command or supervisors.
(2) Record of performance, acts, conduct, or negligence during the period concerned.
(3) Correspondence from creditors, attempting to collect a debt from the Soldier.
(4) Adverse correspondence from civil authorities.
c. Documents will establish a pattern of inefficiency rather than identify a specific incident.
d. Reduction for inefficiency will not be used for the following:
(1) To reduce Soldiers for actions for which they have been acquitted because of court-martial proceedings or civil proceedings.
(2) In lieu of UCMJ, Article 15.
(3) To reduce a Soldier for a single act of misconduct.
e. The CDR reducing the Soldier will inform him or her, in writing, of the action contemplated and the reasons. The Soldier will acknowledge receipt of the memorandum by endorsement and may submit any pertinent matter in rebuttal.
Any matter submitted by the Soldier must be considered by the reduction board and reduction authority prior to rendering a decision.