The higher offer was immediately rejected by Cook and drew a mixed reaction from the stock market, with one big shareholder saying it was "not obviously a knock-out".
Triplex shares dropped 11p to 192.5p, while William Cook's gained 5p to 380p. At that level, Cook's shares still remain 9p below the new bid terms of 21 new shares plus pounds 37.40 in cash for every 20 in Cook. There is a cash alternative of 383.4p a share.
Colin Cooke, the Triplex chairman, said they had decided to raise the offer in line with Cook's original profit forecast of pounds 10.7m for the current year to March, increased over the weekend to pounds 11m. The group has also raised its forecast savings from pounds 1.5m to pounds 2.5m.
The new bid represented an exit price-earnings ratio of 10.8, which is exactly the same as the original terms -"reasonable, [but] not over the top", Mr Cooke claimed.
They had looked at the forecast and it "appears sensible", he said. The difference between analysts' expectations of around pounds 8m at the time the original offer was made in November and the current forecast related to an apparent reversal of the fall in order books revealed at the interim stage and the decision to cut refurbishment costs in the second half.
But he hit out at claims by Andrew Cook, the William Cook chairman, that the new bid still "massively undervalues" the company. "From buying in shares and saying there is no future for the company, now we are seeing sunshine in the air and daffodils in the garden... it is time shareholders got some fair treatment from Mr Cook."
The Cook chairman said the new offer "may no longer be ludicrously low; but it remains manifestly low". He claimed it was worth 10.5 times forecast earnings, a 31 per cent discount to the market, or 8.7 times if pounds 2.2m of refurbishment costs were added back to the pounds 11m forecast.
One of William Cook's big institutional shareholders last night lent support to that argument, saying: "Superficially [the new bid] doesn't look all that impressive.
"There will be a lot of shouting to come, but it's not obviously a knock- out."