Andy Carroll put the east Londoners in front at the KCOM Stadium but Andrew Robertson's fine finish drew the hosts level.
And West Ham eventually succumbed to late pressure as Andrea Ranocchia scored his first goal for the Tigers to boost Marco Silva's survival hopes.
Bilic's side are now only six points above the relegation zone with eight games left to play and the Croatian manager is expecting discussions regarding his tenure to remain on the agenda.
"The only way you can stop the speculation is if you win games or get a point or climb up the table," Bilic insisted.
"After four defeats of course I can expect even more speculation, but as I said I wasn't worried before and I'm not worried now.
"I'm doing my job. I'm very motivated and very positive.
"Speculation and all that - I said ages ago it doesn't affect me. I don't read it. I know it, it's there of course.
"We are all disappointed, everybody with me in the dressing room, but my morale is always high - it's my job.
"I'm totally motivated and focused. We have another game very soon, a chance to make it up, to start winning games."