The men, low-wage workers from India's poorest states, have been stuck in the 4.5km tunnel in Uttarakhand state since it collapsed on November 12.
So-called rat miners, brought in on Monday to drill through the rocks and gravel by hand from inside a 90cm-wide evacuation pipe that has been pushed through the debris after machinery failed, made good progress overnight, officials said.
"Earlier, we ran into metal obstacles, but that has reduced now," Uttarakhand chief minister Pushkar Singh Dhami told reporters. "Now we are finding more concrete which is being cut with cutters."
"We are hoping that the way in which the work is making progress there will be a breakthrough very soon," he said. "Once the pipe goes through the process of removing the men will be initiated."
The men have been getting food, water, light, oxygen and medicines through a pipe but efforts to dig a tunnel to reach and rescue them with drilling machines have been frustrated by a series of snags.
Rescuers on Monday brought in the "rat miners", experts at a primitive, hazardous and controversial method used mostly to get at coal deposits through narrow passages. Their name comes from their resemblance to burrowing rats.
One of the miners was shovelling dark grey debris of rocks and broken concrete with a hand-held spade-like tool inside the dimly-lit, narrow pipe, a video clip shared by authorities showed.
A small group of local women sang Hindu devotional songs outside the tunnel, praying for the rescue to succeed.
"We ran into metal rods and big chunks of concrete and we removed it," Rajesh Kumar, one of the miners, told Reuters. "We hope this will be completed soon. The rest is up to God."
Relatives of some of the 41 men who have been camping near the site have been told to keep the clothes and bags of workers ready to accompany them to a hospital about 30km away after they are brought out, one official said.
The tunnel is part of the $US1.5 billion ($NZ2.4 billion) Char Dham highway, one of Prime Minister Narendra Modi's most ambitious projects, aimed at connecting four Hindu pilgrimage sites through an 890km network of roads.
Authorities have not said what caused the cave-in but the region is prone to landslides, earthquakes and floods.