Herod's temple had court of women. Where did Hannah go to pray in the tabernacle in Shiloh?
In the wilderness, the tabernacle of mobile. By the time of Eli, it was a bit different. It was called a house and there were doors.
1 Samuel 3:15 Samuel lay down until morning and then opened the doors of the house of the Lord.
As the tabernacle remained stationary at Shiloh for 300 years, naturally numerous buildings of a more solid nature grew up around it.
There was a special seat located on a kind of a porch in front of God's house.
Cambridge Bible for Schools and Colleges
upon a seat by a post of the temple] Rather, “upon the seat (or, his seat) by the doorpost of the temple.” “The sanctuary itself was so encased with buildings, as to give it the name and appearance of ‘a house’ or ‘temple.’ There was a gateway with a seat inside the doorposts or pillars which supported it. It was ‘the seat’ or ‘throne’ of the ruler or judge, as afterwards in the Palace of Solomon. Here Eli sat on days of religious or political solemnity, and surveyed the worshippers as they came up the eminence on which the sanctuary was placed.” Stanley’s Lectures on the Jewish Church, I. 321.
Eli could sit there and observed worshippers in the open court.
Matthew Poole's Commentary
And although this tabernacle was but a tent, yet it was supported by boards and posts, and especially at the entrance, by which Eli sat, even by the entrance into the outward court, otherwise he could not have seen Hannah.
1 Samuel 1:13 Hannah was praying in her heart, and her lips were moving but her voice was not heard. Eli thought she was drunk.
Eli sat on a porch with the tabernacle tent housed behind him. He looked outward to the open court area and saw Hannah's lips praying toward the direction of the Holy of the Holies.