(CNN) -- New Hampshire's Senate has approved a bill allowing same-sex couples to marry, but critics and supporters of the legislation say their work isn't over.
New Hampshire could become the fifth state to allow same-sex marriage if a bill becomes law.
"We were obviously disappointed," Kevin Smith, executive director of the conservative Cornerstone Policy Research Action group, said Thursday. "We don't think the voters are going to forget about it."
On the other side, members of the New Hampshire Freedom to Marry Coalition, a group that has worked since 2001 for same-sex marriage, were "absolutely thrilled," said Mo Baxley, its executive director.
But noting that the bill is returning to the House of Representatives for consideration of changes made by the Senate, Baxley added, "I think the work continues." Her organization will encourage supporters to contact their representatives ahead of the House vote, she said.
On Wednesday, the Senate voted 13-11 in favor of the bill, which differs from the House-approved version in that it distinguishes between civil and religious marriage. It allows each religion to decide whether to acknowledge same-sex marriage but extends the option of civil marriage to any two individuals, said state Senate spokeswoman Anne Saunders.
The House, which passed the earlier version last month by a margin of seven votes, 186-179, must approve the changed version before it can be sent to Gov. John Lynch, a Democrat who has questioned the need for the bill.
After the Senate vote, Lynch released a statement saying he believes that "the fundamental issue is about providing the same rights and protections to same-sex couples as are available to heterosexual couples. This was accomplished through the passage of the civil unions law two years ago."
Lynch signed the state's civil unions law in May 2007.
"To achieve further real progress," he added, "the federal government would need to take action to recognize New Hampshire civil unions."
Earlier this month, the governor said he believes the word "marriage" should be used only to describe a marriage between a man and a woman, the New Hampshire Union Leader in Manchester reported.
"I think the word 'marriage' is reserved for a marriage between a man and a woman, and I think the real issues really are rights and protections for gay and lesbian couples," he told reporters on April 15, according to the newspaper.
Smith said his organization, which was established in 2000, would "lobby hard" to get the governor to veto the bill. He said he expected it to pass the House.
"Look, this is a matter of holding the governor accountable," he said. "He's been very public with his views on same-sex marriage in the state. ... We'll see if he was misleading the voters of New Hampshire or if he'll stand by his word and actually veto it."
Were the bill to become law, New Hampshire would become the fifth state allowing same-sex marriage, joining Connecticut, Vermont, Massachusetts and Iowa.
Only Vermont has established the practice legislatively.
CNN's Taylor Gandossy contributed to this report.