The Brussels-based International Crisis Group (ICG) also warned against calls for Indonesia to mimic Singapore's harsh internal security measures with a new intelligence bill due for debate this year.
"The bottom line is that the government is taking terrorism seriously, but much more needs to be done for political will to translate into effective action," the ICG said in a report entitled "Jihadi Surprise in Aceh".
"The Indonesian police, at both local and national levels, deserve credit for their fast work. But no one should be complacent that the job is over."
Police have arrested 48 alleged terrorists and killed another eight, including Dulmatin, one of the region's most wanted men and a key figure in the 2002 Bali bombings, since raiding a terrorist training camp in Aceh province, Sumatra island, in February.
The new group, dubbed "Al-Qaeda Indonesia in Aceh", represented a "third wave" of jihadist extremism in the world's most populous Muslim-majority country, the ICG said.
Its members, several of whom are still on the run, were angry at the perceived softness of regional terror network Jemaah Islamiyah and critical of the tactics of faction leader Noordin Mohammed Top, who was killed last year.
But the death of leaders like Dulmatin and Noordin did not mean the threat had passed, the ICG said.
"Jihadi groups do not disappear after waves of arrests; they evolve and mutate, taking on new forms. The killing of a Noordin here or a Dulmatin there does not eliminate the ideology of salafi jihadism," it said.
"In fact the perceived 'martyrdom' of a few leaders can give the movement new life. It is essential to understand far better how, why and among whom the ideology has taken root and spread in Indonesia."
The government needed to reflect on a series of failures exposed by the extremists' recent activities, including poor intelligence and corruption.
"For all the progress that has been made in the last decade in understanding extremist networks and sharing information across the region about them, the ability to detect their activities remains weak," it said.
"That Dulmatin, one of the region's most wanted terrorists, could leave the Philippines, arrive in Indonesia and live in Jakarta for at least two years without anyone being the wiser suggests that there is still some way to go in improving basic information gathering and analysis."
Dulmatin, a bomb-making expert and one of the masterminds of the 2002 Bali bombings which killed more than 200 people, had "no difficulty getting a fake local identification card and passport," it said.
The Aceh group had also used "corrupt police contacts to buy supposedly destroyed weapons... (illustrating) yet again how corruption becomes the lubricant for terrorist activities".
At least 12 members of the group had criminal records, mostly for terrorism, but had been free to accept visits from other extremists and use multiple mobile phones while in jail.
Terrorists were also using forging syndicates "with ties to government departments", the ICG said.
Despite such glaring failures, the ICG warned against extending arrest and detention powers to agencies other than the police.
"Indonesia's much-vaunted 'soft approach' toward terrorism should not be discarded. It is far more appropriate than a harsher regime, and the last thing Indonesia needs is a Singapore-style Internal Security Act... " it said.