I never noticed it, but there were a lot of milestones achieved for women in the 1860s and 70s. From my knowledge, there was generally an increase in divorce, women on a proper payroll, and overall women were not expected nearly as much to perform household duties. In 1866, Susan B. Anthony and Elizabeth C. Stanton formed the American Equal Rights Association. In the same year, Lucy Hobbs Taylor became the first American woman to receive a dentistry degree. 1872, Victoria Woodhull was legally allowed to run for president (even if she lost miserably). 1878, a women's suffrage amendment was introduced to Congress. 1879, Belva Lockwood became the first American woman to attain a degree in architecture. I'm sure there are more examples, but you get the idea. Was it the relatively large amount of deaths of the Civil War? Something else about the Civil War? A coincidence? Something else entirely?
It's not a coincidence, but it's more complex than a cause -> effect in either direction.
When we talk about the earliest history of the women's rights movement, we often pick out individuals who stood up for women's abilities and said "things could be arranged better than they are" throughout history, like Christine de Pisan, Mary Wollstonecraft, Olympe des Gouges, etc. As an organized political movement, however, there is a pretty defined start date: 1848. This was the year of the Women's Rights Convention at Seneca Falls, NY, organized by Elizabeth Cady Stanton, Mary Ann M'Clintock, Lucretia Mott, and others.
I think it would be fair to say that the post-French Revolution west was an era of reform, or of agitation for reform. Organized support for abolition had been ongoing since the end of the American Revolution, Britons were protesting the outdated electoral system of the UK which deprived industrial areas of representation, the laws that made Catholics second-class citizens in England were being replaced, and so on. A wave of revolutions went across Europe in 1848. It's unsurprising that the women who supported their rights to education and suffrage would also coalesce into an organized movement at the same time, particularly as many of them were familiar with abolitionism, if not active abolitionist reformers themselves: a large proportion were Quakers, and the sect had been staunchly abolitionist for many years; Stanton was also closely related to Gerrit Smith, one of the leading abolitionists of the time. Given this, it's also unsurprising that once the abolition of slavery had been achieved, they would come to focus more energy into making gains for women's rights - both because an amendment banning slavery was a massive win, and because many of them were racists who wanted to achieve the concrete goal of abolition but did not feel the need to continue campaigning for the rights of African-Americans. This led to a narrowing of focus and successive improvements and milestones.
However, plenty of milestones relating to the movement or women's rights outside of it occurred before the war. Acts to protect the property of married women and the vulnerability of women to their husbands' debts were being passed by a number of states before and after 1848. Elizabeth Blackwell was the first woman to be accepted to and graduate from medical school (in the 1840s), and she would go on to found an infirmary in New York City in the 1850s, to compare to Lockwood and Taylor. However, yes, more happened after the war. Time is a major factor here: as an organization grows and ages, unless it falls into decline it will continue to make progress, and since the movement was organized only a little over a decade before the Civil War and continued to exist for many decades after it, it's inevitable that most of its successes would come during and after Reconstruction.
For more on the history of the women's rights movement, I cannot recommend enough Laurel Thatcher Ulrich's Well-Behaved Women Seldom Make History.