The 22-year-old was met with obstacles at each step - from a real estate agent demanding cash for every viewing, despite it being illegal, to a landlord who asked for 18 months' rent in advance, instead of the standard 12.
"I found a two-bedroom [apartment] for 950,400 naira ($2,640) per annum. The rooms were so tiny they could barely fit three people at once," Odili told Al Jazeera.
Lagos is home to 22 million people and counting, more than double New York and London's tally.
The city's population grows by 77 people every hour as Nigerians from less industrialised regions seek jobs. And as the city grows, so too does demand for housing.
In a country where the minimum wage is about $80 a month and where graduates earn an average of 80,000 naira ($222) a month, renting in Lagos is an expensive exercise.
Odili got lucky. Her employer offered her a housing loan and she was ready to move in with a friend.
But just as she secured a place, she received a call from the agent saying the property had been rented out to someone else.
"After that call, I made up my mind to stop looking."
Gender discrimination is also at play in the search for a home. The odds are stacked against the young, unmarried woman like Odili.
Women living outside their fathers' or husbands' houses is considered inappropriate in some sections of Nigerian society, so single women looking to rent a property are often rejected or subjected to more rigorous screenings.
The housing shortage is also exacerbated by unoccupied luxury apartments in wealthy Lagos suburbs, including Ikoyi and Victoria Island where rent typically begins from $20,000 a year, in a country where almost half of its citizens live on less than $2 daily - a problem which has been acknowledged by Nigeria's Minister for Works, Power and Housing, Babatunde Fashola, a former Lagos state governor.
With little government housing, the onus falls on the private sector.
An estimated two-thirds of people in the Lagos metropolis live in informal accommodation or slums. Some jostle for space in crowded shanties, often built on stilts in water communities where residents live under the threat of eviction or in dilapidated buildings prone to collapse.
Proposed solutions from mega-projects to start-ups
The Eko Atlantic project, which began in 2009, aims to solve the dual problem of job and housing shortages in Lagos.
It is currently being built on 10 million square metres of land reclaimed from the Atlantic Ocean and protected from rising sea tides by an 8.5km wall dubbed the Great Wall of Lagos.
Once complete, it will comprise housing blocks for 250,000 people, a financial district, and private water and power.
The project envisions itself to be the size of Manhattan's skyscraper district and there have been suggestions it will only cater to Lagos's ultrarich, a tag developers are trying to shake.
Nnimmo Bassey, an environmentalist and activist, describes the project as "climate change apartheid", telling Al Jazeera it further deepens the economic divide "between the filthy rich and the rest of the people struggling for survival".
There are concerns over the project's potential effect on surrounding communities, particularly how the low-lying areas will cope in the event of a storm surge.
Bassey believes claiming marine land has destroyed ecosystems and could lead to the extinction of some species.
"It swallowed up public beaches and deflected coastal erosion to other communities. Some communities are getting hit by coastal erosion pushed their way by this project and they require heightened levels of investment to secure them."
Ugochi Oluigbo, an environmental journalist, says Eko Atlantic has not been well received in neighbouring areas.
"I have been to some communities along the Atlantic in Lagos to cover incidents of flooding and usually, they point to the big project far away, and blame it for their woes," Oluigbo told Al Jazeera.
We had to solve the affordability problem because, in Nigeria and most parts of Africa, there is a rental price and income mismatch.
Lagos-based start-up Muster, meanwhile, is attempting to improve on renting conditions based on "affordability, availability, convenience and flexibility", says Ibraheem Babalola, who runs the company.Muster allows would-be renters to connect with homeowners and property managers.
They can rent for a minimum of three months, pay monthly, quarterly or annually depending on their preferences, and find people of their choosing to share apartments with.
Those with spare rooms to let can also list on Muster, which allows both parties - renters and owners - to skip legal and agent fees.
"We had to solve the affordability problem because, in Nigeria and most parts of Africa, there is a rental price and income mismatch. On average, people aged between 20 and 34 years old earn around $230 monthly, which is lower than the average price of renting a one-bedroom apartment in the major cities in Nigeria," Babalola said.
Left: The exterior of an old shipping container as it is being converted into an office building in Lagos.
Right: A completed office building made out of shipping container [Courtesy of TempoHousing Nigeria]
"For a long time, property developers were building with family units in mind with three-five beds built. Young people who only needed smaller spaces couldn't find fit-for-purpose apartments," he said.
Beyond providing affordable and flexible housing alternatives, Muster attempts to tackle gender discrimination.
"A lot of traditional landlords ask to see women's marriage certificate or husband before renting [out] to them but nobody will ever ask that question on Muster. It will never become a requirement," said Babalola.
TempoHousing, another Lagos-based start-up, is even more revolutionary in its attempt to make housing affordable.
It converts cargo containers into spaces fit for living and aims to "challenge the stigma attached with alternative construction," said Dele Ijaiya-Oladipo, a managing partner at the company.
Ijaiya-Oladipo says container homes are 30 percent cheaper than traditional homes and can be easily transported. Construction can take as less as 10 days.
But despite what seems like obvious advantages, there's some scepticism among Nigerians about container homes due to its novelty and the social status associated with living in traditional houses.
"There was a lot of initial scepticism towards the product due to a precedent not being set. However, over the years, we have actively tried to change this by showcasing as much as possible," he said.
As for Odili, whose search came to an abrupt end in Lagos, she has found a trusted and traditional solution.
"I am going to stay with my parents for as long as possible. The experience stressed me out," she said.