The council laid out plans to achieve an interim target by 2035.
The retirement fund liquidated a $243 million international equity mandate and made commitments totaling $770 million in December.
The system will consider making a growth private equity commitment at its Feb. 1 board meeting this month.
The system is searching for a director to help oversee and manage its portfolio with a focus on private equity.
The fund approved private equity and private credit commitments with existing managers as recommended by its general investment consultant last week, with more recommendations expected in the coming months.
Fund was oversubscribed and closed above its initial hard cap of $6 billion.
The plan is expected to issue a private equity manager RFP in the next few months.
The commitments include four new managers for the plan.
The Lone Star pension system added a pair of private equity commitments totaling $45 million in recent months.
The six commitments represent new relationships for the pension plan.