Collins also nodded wide from a decent position and Kouyate, the match victor here against Swansea City in the Hammers' last home league outing, miscued a shot wide from 25 yards out.
Everton's Romelu Lukaku, left and West Ham's Winston Reid battle for the ball, during the English Premier League soccer match between West Ham United and Everton, at London Stadium, in London, Saturday April 22, 2017.
Ireland global Randolph, 29, was installed as Bilic's first choice in November - but he took stick for mistakes at the Stadium of Light.
Everton, meanwhile, surely waved goodbye to the Champions League dream in the meekest way possible.
Neither side was able to find a way to break the deadlock with Everton striker Romelu Lukaku failing to extend his run of having scored in his nine previous games against the Hammers.
Nevertheless, it was the Hammers who started the brighter, with both Jonathan Calleri and Cheikhou Kouyate a whisker away from a telling contact, this after pinpoint delivery from Edimilson Fernandes and Manuel Lanzini.
It bounced off his knee across the unguarded penalty area but Adrian managed to recover in the nick of time, smothering the ball as Kevin Mirallas closed in on what would have been a simple tap-in!
Trump asks for probe into imports of foreign-made steel
That would force up their prices and make domestic steel prices competitive, enabling closed mills to reopen and rehire. By law the investigation must be concluded and a report submitted within 270 days.
One fan stated that Calleri starting for the London-based club proves how bad it now is at West Ham.
"If someone is interested, for sure they will call the board and then we will speak about it but we will try to do everything to keep him at Everton next season".
"Yes, [Reid's] back from injury, which is good for us", Bilic said in his press conference. He's been scoring goals all the time.
He sent on Gareth Barrie and Ademola Lookman in the hope they could inject something into his Everton side but even that failed. Indeed, a long-range attempt from Lookman was all the Toffees had to show for their toil as the half reached its midway point.
West Ham have been extremely disappointing since the start of March.
The match remained in the balance going into the closing stages, Lanzini cutting inside from the left and testing Stekelenburg from 25 yards, while Lookman's dipping strike had Adrian anxious at the other end before flying just over.
The result sees Everton (16-8-10) remain in seventh place on 58 points, while West Ham (10-16-8) stay in 13th on 38 points.
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