Try these:
Hat Tricks
61, Hindburn Close, Doncaster, South Yorkshire DN4 7RW
Tel: 01302 533505
Pat's Hats (Hat Hire)
Millfield Cottage, Gainsborough Rd, Everton, Doncaster, South Yorkshire DN10 5BW
Tel: 01777 817449
Jacs Hats for Hire
39, Leadbeater Rd, Sheffield, South Yorkshire S12 2UG
Tel: 0114 239 3737
Cappelli Hat Hire
25, Rippon Rd, Sheffield, South Yorkshire S6 4ND
Tel: 0114 233 5796
Hire-a-Hat
228, Urban Rd, Hexthorpe, Doncaster, South Yorkshire DN4 0HJ
Tel: 01302 85253
Bonnets Hat Hire
28, Vernon Delph, Sheffield, South Yorkshire S10 5NS
Tel: 07974 994413
Felicity Hat Hire
158, Baslow Rd, Totley, Sheffield, South Yorkshire S17 4DR
Tel: 0114 262 0530
All of the above have been taken from the listings on Yell.com. I've only included firms where the name or advertisement makes it clear that they hire hats as well as (or instead of) selling them. If, however, you look under 'Hat Shops & Milliners' in your local Yellow Pages you may well find additional firms. (This is because many firms take out descriptive advertisements in Yellow Pages but not on Yell.com).
Chris