The traditional story of the Timmer Hat is a popular tale often told to relate William leaving Bello Mill and walking the 250 miles to Birmingham to seek employment with Boulton & Watt. 

For the journey, the tale tells of William making himself a "Timmer" or top hat turned from a single piece of wood on a foot operated oval turning lathe of his own making. This is one of the earliest recorded uses of a lathe. 

Suitably attired with his new "Timmer" hat, William set off on the walk to the Soho Foundry in Birmingham where James Watt was developing his steam engine for the industrialist Mathew Boulton. 

On arrival at Soho, he was interviewed by Boulton as Watt was absent from the works. 

REFERENCES :
Samuel Smiles : Iconographer & Biographer, writes:

During the brief conversation that took place, the blate [diffident] young Scotchman, like most country lads in the presence of strangers, had some diffculty in knowing what to do with this hands, and unconsciously kept twirling his hat with them. Boulton's attention was attracted to the twirling hat, which seemed to be of a peculiar make. It was not a felt hat, nor a cloth hat, nor a glazed hat : but it seemed to painted, and compsed of some unusual material. 

"That seems to be a curious sort of hat," said Boulton, looking at it more closely; "what is it made of?" 
"Timmer, sir" said Murdoch modestly. 
"Timmer? Do you mean it is made of wood?" 
"Deed it is, sir" 
"And pray how was it made?" 
"I made it mysel, sir, on a bit laithey of my contriving." 
"Indeed!" 

Boulton looked at the young man again. He had risen a hundred degrees in his estimation. William was a good looking fellow - tall, strong, and handsome - with an open intelligent countenance. Besides, he had been able to turn a hat for himself with a lathe of hs construction. This, of itself, was a sufficient proof that he was a mechanic of no mean skill 

"Well!" said Boulton, at last, "I will enquire at the works, and see if there is anything we can set you too". 
 

A more likely scenario is that he set off with a more specific purpose in mind. 

Probably when he arrived at the Boulton & Watt Manufactory, enquiry showing Mr Watt to be absent, William sent a verbal or written message to Boulton to the effect that he was a skilled millwright, known to Mr Boswell (and Mr Watt?), and would be obliged if Mr Boulton considered him for a post. Boulton would have recollected Boswell's remarks about Murdoch and agreed to see him : 

Boulton would have seen a young man of apparent great strength not just of body but of character, demonstrated not least in his walking the 250 miles to Soho. 

Boulton : "I am afraid I have nothing for you young man, I am at present putting men off, not taking them on."

Murdoch : "Would there be nothing sir, on the steam-engine side? I would dearly love to work with Mr Watt on that matter."

Boulton : "I am, as it happens, looking to train engineers to erect steam-engines but that is skilled work. What can you do in that way?"

Murdoch : "I understand pattern making, sir, and casting, I have good arithmetic and understand mills and tools. I devised my own lathe and on it I turned this hat [He lays the wooden hat on Mr Boulton's table.]

Boulton : "This is turned as a single piece?"

Murdoch : "It is, sir."

Boulton : "There are few, if any, I know with the skill to do that. I will enquire at the works and see if there is anything we can set you to."

Murdoch : "Thank you Mr Boulton, sir, you will not regret it. I shall serve you well."