“A large majority of retailers continue to express concerns that they have no forward visibility of their next deliveries,”, he said.
He added: “There are many reports of wet sites quickly going dry because the continuity of tankers remains out of kilter with orders.
“The situation in London and the South East remains serious.”
Mr Madderson said there had been “a welcome improvement” in the region over the weekend with 10% of non-motorway sites running out of fuel, which was “not far behind the rest of the country”.
Watch: Fuel crisis: Petrol prices continue to rise as cost on forecourt reaches highest point since 2013 and shortages remain serious
The figure was based on a survey of a quarter of petrol stations.
He cited figures from motoring research charity the RAC Foundation which indicated that filling stations in London and the South East were used more intensely than those elsewhere.
Mr Madderson said: “These filling stations therefore need to be refuelled more often.
“The need to refuel filling stations in London and the South East is even more necessary when customers panic buy, because there are more cars to be filled per station there than the GB average.”
The fuel crisis begin when demand surged after BP announced on September 23 it would have to close a handful of its petrol stations.
The Government says it has taken a number of steps to relieve pressure on petrol stations, such as deploying military tanker drivers and exempting the fuel industry from competition laws.
New figures from the Department for Business, Energy and Industrial Strategy show the average price of petrol at UK forecourts was 137.17p per litre on Monday, up 2p per litre from before the crisis began.
Diesel has increased by 3p per litre over the same period, to 140.66p per litre.
These are the most expensive average prices for petrol and diesel since September 2013.
Watch: Energy crisis: Chancellor appears to rule out helping businesses with gas prices