But a growing number are defying the law and appearing bareheaded in the streets.
The trend accelerated during the nationwide protests sparked by the September death in custody of Mahsa Amini, a 22-year-old Iranian Kurdish woman arrested for allegedly violating the law.
The protests rocked Iran, provoking a crackdown by authorities that claimed the lives of hundreds of people, including dozens of security personnel, and saw thousands more arrested.
Iran's conservatives, who dominate the country's parliament and leadership, have passionately defended the dress code and believe relaxing rules would begin a process leading to profound shifts in "social norms".
But with many Iranians demanding change, in May the judiciary and the government proposed a "Support for the Culture of Hijab and Chastity" bill, to "protect society" and "strengthen family life".
The text proposes increased fines for "any person removing their veil in public places or on the internet" but withdraws the threat of a prison sentence.
"This bill reduces the removal of the hijab from a felony to a misdemeanor, similar to a traffic violation but with heavier fines," sociologist Abbas Abdi told AFP.
After Amini's death and the subsequent protests, society "no longer accepts that we imprison a woman because she does not wear the veil", he said.
Since the protests, authorities have imposed a series of measures to enforce Iran's strict dress code, including the closure of businesses whose staff do not conform with the rules and installing cameras in public places to track down offenders.
In recent days, at least three officials have been sacked or arrested for failing to prevent unveiled women from entering historic sites.
- 'Not dissuasive enough' -
Under the proposed law, the text of which has been published in government-affiliated media, offenders will first receive a warning text message from the police.
A second breach will incur fines of between five million and 60 million rials (around $10 to $120), a large sum for many Iranians. The law would also provide for other penalties, including the confiscation of a woman's vehicle for up to 10 days.
Defending the bill, judiciary chief Gholamhossein Mohseni Ejei stressed the need to avoid polarizing society, saying he understood the "concerns of believers" supportive of the dress code.
As the bill awaits examination by lawmakers, it faces accusations of not being tough enough from ultra-conservatives, an influential bloc in the current parliament.
Relaxing punishments for violations will see "the expansion of a repugnant phenomenon" by "removing legal barriers" for women not wearing a veil, the ultra-conservative newspaper Kayhan said.
Those supporting the law "do not know that the enemy" seeks to "destroy the family as an institution and ultimately, to attack the foundations of the Islamic system" by removing headscarves, the newspaper said.
Social networks and foreign media, particularly television channels broadcasting in Persian, are calling for "social disobedience", according to some ultra-conservatives.
Within Iran's leadership "there is no consensus on the hijab", as some favor repression, while others "believe that other means must be tried", the sociologist Abdi said.
"The bill satisfies neither the supporters of compulsory hijab nor, of course, the supporters of the freedom to cover up or not."
A similar situation developed in the 1990s with a law prohibiting the use of satellite dishes, he said.
"It was only implemented for a while before it was dropped."