I occasionally run into Kidderminster for the agency I’m with in Kent. I would suggest that hourly rates around there should be equal to ours around here, that is £10ph weekday shifts, more at weekends on PAYE basis.
I’m also one of the apparently few drivers who actively values free time over working all hours, so I can afford to be fussy in what I pick up. Two or Three decent length shifts (12-15 hours) does me, but there’s plenty of times I’ll find myself with nothing one week, and doing a full cover 55 hours for someone on leave the following week to average it all out.
If you want part time hours week in, week out, then you’d have to be prepared to do supermarket work - at least around here.
Redditch seems to have a few agency drivers running around on their local depot runs, but I don’t recall much about Kidderminster running in there with a 12tonner.
As to the “availability of jobs & drivers” issue, I’d answer it like this:-
There are a glut of zero hours singleton shifts around the country, and not enough agency drivers to cover them, because they are at depots spread far and wide, perhaps paying too little to be worth the while of a driver commuting in from 50+ miles away.
There are on the other hand a glut of drivers seeking 40+ hours a week employment.
Add to this, a reluctance it seems for agency drivers to find themselves ‘living and dying at the supermarket RDC’ as well, and you’ll come to the same conclusion as I did, that it’s better to keep yourself for the work that makes you a profit than chase the “Loss leader” stuff like low-pay at distance. If you start down that road, that’ll be all you’ll ever get, because the agency will see you coming.
I think If I were living in your neighbourhood, I’d take £10ph in Redditch, Worcester, and other towns west and southwest of B. Inside Brum itself, I’d expect more if beyond the centre.
Inner city work usually does pay more though, especially at short notice. If it doesn’t, find another agency. Don’t listen to those on here that think all drivers are mearly serfs to those that own all the businesses and buy over the politicians to leave them alone to their race-to-the-bottom devices!

Supermarket work I believe pays around £9.00ph base. You’ll need to do lates/nights to get the tenner there. You’ll also want a lot more if pushed onto umbrella/self-employed as well, to make up for lack of holiday pay and having to pay the extra employer NI contributions and umbrella fees etc. I’m quoting reasonable rates for PAYE work. 
I assume in all this of course that you hold a C+E licence?