Top seed Marin Cilic fired another warning to his Wimbledon rivals by easing into the semi-finals at Queen's Club.
The 28-year-old edged a competitive affair 7-6 (7/4) 7-6 (7/4) against Willis at Southlands College near the All England Club.
The match with Findel-Hawkins represents another all-British contest. Evans has missed out on a Wimbledon wild card, and is in the early stages of his return to tennis after being banned last year for a doping offence.
Nick Kyrgios continued his impressive week at Queen's by dethroning 2017 champion Feliciano Lopez.
The Australian won a close contest 7-6 (7/5) 7-6 (7/3) to book his place in a Fever-Tree Championships semi-final against top seed Marin Cilic.
Kyrgios, who has beaten Andy Murray and Kyle Edmund this week, has the wood over Lopez, having also got the better of him in Stuttgart last week.
He served 32 aces again, as he did in beating Edmund yesterday, matching his personal best.
Novak Djokovic became just the 10th man in the open era to register 800 victories.
The former world number one achieved the feat when he beat Frenchman Adrian Mannarino 7-5 6-1 to continue his ominously good run of form.
Djokovic, who beat Grigor Dimitrov, has qualified for just his second semi-final since Eastbourne last June.
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