I need to solve a cylindrical diffusion problem that is defined in $[1,\infty]$. I would like to use Hankel Transform that has is defined on $[0,\infty]$. So in order to apply Hankel transform in my case, do I need to change the Kernel of the Transformation.
Yes, you do.
In general, the Bessel solutions to the boundary value problem require that the function remains finite at r = 0.
If your domain does not include r = 0 (and yours does not!) then you need to use a linear combination of Bessel functions of the first, and second kind.
A bit of Googling will help you to get a good idea about how to go about this. For example, "Theory and Operational Rules for the Discrete Hankel Transform" is a very nice paper which attempts to put the DHT on equal footing with the discrete Fourier transform.