With regard to 1 Corinthians 7:36 the King James Version uses the word “virgin” but the expression “the virgin he is engaged to” (as expressed in the NIV) is nowhere to be found in manuscripts. The ESV uses the word “betrothed” and the NLT says “fiancé”. Here is some information about being engaged (or betrothed) and being married:
The origins of European engagement in marriage practice is found in the Jewish law (Torah), first exemplified by Abraham, and outlined in the last Talmudic tractate of the Nashim (Women) order, where marriage consists of two separate acts, called erusin (or kiddushin, meaning sanctification), which is the betrothal ceremony, and nissu'in or chupah, the actual ceremony for the marriage. To be precise, nesiuin is the process, and chuppah is the method. This was later adopted in Ancient Greece as the gamos and engeysis rituals, although unlike in Judaism the contract made in front of witness was only verbal. The giving of a ring was eventually borrowed from Judaism by Roman marriage law, with the fiancé presenting it after swearing the oath of marriage intent, and presenting of the gifts at the engagement party. Source: https://en.wikipedia.org/wiki/Engagement#Origin
In Jewish weddings during Talmudic times (c.1st century BC – 6th century AD), the two ceremonies of betrothal (erusin) and wedding usually took place up to a year apart; the bride lived with her parents until the actual marriage ceremony (nissuin), which would take place in a room or tent that the groom had set up for her. Source: https://en.wikipedia.org/wiki/Engagement#Betrothal
Here is some information on how the Greeks viewed marriage:
Women in Classical Athens had no legal personhood and were assumed to be part of the oikos (household) headed by the male kyrios (master). Until marriage, women were under the guardianship of their fathers or other male relatives; once married, the husband became a woman's kyrios. While the average age to get married for men was around 30, the average age for women was 14. This system was implemented as a way to ensure that girls were still virgins when they wed; it also made it possible for husbands to choose who their wife's next husband was going to be before he died. Source: https://en.wikipedia.org/wiki/Women_in_Greece
The truth about sex in ancient Greece by James Robson, Senior Lecturer in Classics, The Open University (U.K.): As for marital relations, men seldom married before the age of 30, and apart from the wedding night, it was common for married couples to sleep apart... A girl’s father traditionally saw it as his duty to find a suitable husband for his daughter and, importantly, would generally have played a role in finding a wife for his son as well. In Athens, a girl generally got married at about 16 – typically to a man twice her age, often a paternal uncle or an associate of her father’s. Source: http://theconversation.com/the-truth-about-sex-in-ancient-greece-39025
The Virgin Goddess (Artemis) NOTE: copyright protected (2003-2016): It’s important to clarify that ‘virginity’ was a very different thing in ancient Greece than it is for us in the modern world. In ancient Greece, Artemis was known as a virgin Goddess. ... To them, sexual abstinence was an after effect of virginity, not the definition of the word... Parthenos is the Greek word that generally gets translated as “virgin”. Determining what it actually meant to be a parthenos is incredibly complicated, and one is likely to come up with a different answer for every polis... So, while what we conclude about ancient Greek virgins based on Athenian social structure might not apply to all city-states (Sparta being the classic example), it will hold true to most of them, and will be at least relevant to all of them.
So what did it mean to be an Athenian parthenos? Ideally it meant an unmarried young woman who still lived with her father and never had sex. Athenian men wanted to ensure that their wives would bear legitimate heirs, so daughters were kept under strict supervision and seclusion to ensure that they had not been exposed to any other men before meeting their husbands. Source: http://www.neokoroi.org/religion/gods/artemis/the-virgin-goddess/
The short answer to your question is that a woman who was not married was expected to be a virgin, and a woman who became engaged to, or betrothed to, a man would remain a virgin until the wedding night. Marriage follows a period of engagement. This appears to be the norm in both Jewish and Greek society during the first century.