What was the name of the halfway house
Pete Rose entered upon leaving prison ?

Talbert House

Here is an ESPM.com article verifying it.

Jan. 7
Baseball
1991: On a cold, gray day, Pete Rose -- No. 14 for the Cincinnati Reds and No. 01832061 at Marion Federal Prison -- is released from the minimum-security facility in Illinois after serving five months for federal income-tax evasion. Later in the day he reports to a Cincinnati halfway house.

Baseball's all-time hits leader with 4,256 will serve his sentence at Talbert House, where he will be confined in the evenings for three months. He will live in a room with a metal spring bed, a steel trunk for belongings, and a bathroom. The residents live two or three to a room. During the day, he will begin his 1,000 hours of community service that are part of his sentence.

After the three months, Rose will continue his community service and be on probation for nine months.

Rose earned 11 cents an hour working eight-hour days in Marion Prison's welding shop.

I remember seeing an interview with Rose after he was released from Marion. He spoke of how hard life in prison was, and that people just didn't understand. One of his complaints (and I am serious here, as he was) was that the prisoners had to walk to another building for their meals. He pointed out that the building was "3 or 400 yards away" and that "some days it was raining." I'm glad he was able to survive such brutal conditions