The deaths came as Israeli strikes killed 41 others, including 11 children, according to hospital officials.
The Gaza Humanitarian Fund (GHF) accused the Hamas militant group of fomenting panic and spreading misinformation that led to the violence, though it provided no evidence to support the claim.
It said on Wednesday that 19 people were trampled in a stampede and one person was fatally stabbed at a hub in the southern Gaza city of Khan Younis.
Gaza's health ministry and witnesses said GHF workers used tear gas against the crowd, inciting a panic.
The ministry said it was the first time people have been killed by a stampede at the aid sites.
It was also the first time that GHF has confirmed deaths at one of its distribution sites, although Palestinian witnesses, health officials and UN agencies say hundreds of people have been killed while heading to the hubs to get food.
Some witnesses said the crowd panicked after receiving messages that no aid would be distributed or would only be distributed later.
Others said people became trapped while attempting to move through a turnstile system, which creates a bottleneck.
The aid sites are inside Israeli military zones protected by private American contractors. Israel troops surround the sites, but the army says they are not in the immediate vicinity.
The United Nations human rights office and Gaza's Health Ministry said on Tuesday that 875 Palestinians in the enclave have been killed while seeking food since May, with 674 of those in the vicinity of aid distribution sites run by GHF.
The ministry and witnesses say most of the deaths have come from Israeli gunfire. The Israeli army says it fires warning shots and only uses live fire if crowds threaten its soldiers.
GHF, an American organisation registered in Delaware, was established in February to distribute aid during the ongoing Gaza humanitarian crisis.
Meanwhile, Israeli strikes killed 22 people in Gaza City, including 11 children and three women, and 19 others in Khan Younis.
The Israeli military said it has struck more than 120 targets in the past 24 hours across the Gaza Strip, including Hamas military infrastructure of tunnels and weapons storage facilities.
Israel blames Hamas for the civilian deaths because the group often operates in residential areas.
Also on Wednesday, the Israeli military announced the opening of a new corridor - the fourth - that bisects Khan Younis, where Israeli troops have seized land in what they said is a pressure tactic against Hamas.
Negotiations in the Qatari capital between Israel and Hamas are at a standstill after 21 months of war, which began with the militants' cross-border attack on October 7, 2023.
That day, militants killed some 1200 people, mostly civilians, and abducting 251 people, most of whom have since been released in ceasefire agreements or other deals.
Fifty hostages are still being held, less than half of them believed to be alive.
Israel's retaliatory offensive has killed more than 58,000 Palestinians, according to Gaza's health ministry, which has said women and children make up more than half of the dead.
