The escalating unrest, which last week sparked petrol shortages that forced the government to dip into strategic fuel reserves, comes less than two weeks before football fans pour into France for the Euro 2016 championships.
President Francois Hollande and the government are refusing to give in to the hardline CGT union's demand that it withdraw the planned reforms.
Air travellers are also set to face more cancellations and delays.
Six of France's eight oil refineries were Sunday still halted or running at reduced capacity due to action by union activists.
Strikes continued at oil terminals in the southern city of Marseille and at the terminal in the northern port of Le Havre, which supplies kerosene to Paris's two main airports. A skeleton service is however allowing some supplies to get through.
The government insisted that the situation would improve on Monday for motorists who have been forced to queue for petrol and have been restricted in how much they can buy in many parts of the country.
"There will be petrol. Things are getting better although we need to stay on our guard," government spokesman Stephane Le Foll told France 3 television.
Riot police on Friday cleared all the remaining blockades at fuel depots, allowing more supplies to reach petrol stations. One depot remains on strike.
Hollande's tough line was echoed by Prime Minister Manuel Valls on Saturday, when he held talks with bosses in the oil and transport sectors.
He pledged to defend the labour law reform "to the end" and said withdrawing it would be a "bad thing for working people".
At issue are measures aimed at injecting more flexibility into France's famously-rigid labour market by making it easier to hire and fire employees.
Companies would also be able to negotiate terms and conditions with their workers rather than be bound by industry-wide agreements.
But the unions say the moves will erode job security and fail to bring down unemployment which is stuck at just under 10 percent.
Forty-six percent of French people want the government to scrap the reforms, according to an opinion poll in the Journal du Dimanche newspaper on Sunday. Forty percent think they should be modified and just 13 percent want them left unchanged.
The conflict comes a year before presidential elections in which Hollande is considering seeking a second term despite popularity ratings that are among the lowest for a post-war French leader.
There is no sign that the unions will back down soon.
They have called for another national day of rallies and strikes on June 14th, the day that the Senate, the upper house of parliament, begins examining the law.
All eyes this week will be on the railways. With negotiations on working conditions delicately balanced, unions have called for rolling strikes to begin on Tuesday.
They look set to be more disruptive than last week when four out of five trains ran despite strikes calls.
Mass stoppages would add to mounting problems for rail operator SNCF which had to deal with power cuts on Saturday and Sunday that crippled part of the network.
Civil aviation unions have called for a strike from next Friday to Saturday over their own demands.
A leading member of the Socialist Party tried to play down concerns that the strikes will play havoc with Euro 2016 which kicks off on June 10th and will take place at 10 venues across France.
"There will be no train or Metro strikes during Euro 2016," Jean-Christophe Cambadelis, the party's secretary general, said.
He said the unions were mistaken "if they believe for a second that they will take France hostage".
But Alain Juppe, the centre-right former prime minister who leads opinion polls for the presidential election, said he was concerned that the unions were digging in.
"This is going to go on for the whole of June and perhaps for the whole of July too," he said on Friday.