Am I correctly showing a hydride shift on the alkyl halide?
When you solvolyze the starting bromide a secondary carbocation would be generated.  Since a secondary carbocation is not particularly stable, the molecule will explore other pathways such as hydrogen shift, alkyl shift, etc., in order to produce a more stable carbocation.  The figure below, explores these options for your starting bromide.  Realize that although I've drawn a discreet secondary carbocation that then undergoes rearrangement, in reality cation generation and alkyl or hydrogen shift may occur in a more or less concerted fashion.



In this case there are 3 options (actually there are 4 options, but the fourth one would produce a primary carbocation, so I left it out - can you find it?), a, b and c. Pathways a and c produce secondary carbocations; however pathway b, involving hydrogen shift, produces a more stable tertiary carbocation.  Capture of the tertiary carbocation from pathway c by solvent should produce the expected product.