Here's some history gleaned from newspapers:
1763 there was a 'Woolpack' on Moor Street
1794, known as the Woolpack Inn
1816, Mrs Budd was at the sign of the Woolpack in Moor Street
1819, known as the Woolpack Inn
Until 1866 known as the Woolpack Inn
1869, known as the Woolpack Hotel
1877 Known as the Woolpack Commercial Hotel (was this a rebuilt Woolpack Inn ?)
Known as the Woolpack Hotel until 1956
1956 the Woolpack Hotel was compulsorily purchased by Birmingham City for Inner Ring Road development and subsequently demolished.
Source: British Newspaper Archive