There is actually case law on this and the answer is quite
An agreement to pay more money can be sufficient consideration if
you derive practical benefit. There must also be no economic duress.
The leading case is Williams v Roffey and Google will give you plenty of reading
however the circumstances are a bit different in that the contractor was
offered more money to get the job completed on time and the extra money that
was offered, was never paid. That is why he sued so it is a bit different than
yours in that in the case I quote they offered money whereas in your case, it
is being demanded.
It is also worth googling Stilk v Myrick which is probably closer
to your scenario.
In your particular case I think you are entitled to rely on the
price that is quoted. In addition, he is demanding money to finish the job
which puts you under pressure.
From a practical point of view, I would agree to pay him what he
is demanding but then don't pay it and let him sue you or, if he will not
finish the work without payment, pay him, wait until the job is finished and
then sue him for the money back on the basis that paid the pressure/duress
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