"after some effort" is correct and can be used in written English.
It is often used to indicate the result of a particular effort or action. Example: After some effort, she was finally able to open the jammed door and escape from the burning building.
After some effort, I managed to ask about a paper I'd read regarding the use of tDCS to treat tinnitus.
After some effort, we found someone who let us see the renderings, as long as we didn't say who it was, and didn't share them with you.
On instructions from Alfonso, the young men say they are going off to war, then reappear disguised as exotic Albanians and, after some effort, successfully seduce each other's betrothed.
It generally appears only after some effort on the part of the helper to understand what is really happening in the business, generally through time spent with the owner.
After some effort, I found Mr. Murphy's windowless office up a winding linoleum staircase in the back of the second floor of Shopper's World, past aisles of elastic slippers, towels and DVDs.
After some effort, he receives a breakdown of the charges; the fund manager gets 1.35%, the adviser who sold him the pension 0.5%, there is a wrap fee of 0.81% and transaction costs (buying and selling shares) of 0.19%.
Nasr, though, went within 0.076s of making it to Q3; some effort after the legal problems the team have had this week.
The Bastille's archives, recording the operation of the prison, had been scattered in the confusion after the seizure; with some effort, the Paris Assembly gathered around 600,000 of them in the following weeks, which form the basis of the modern archive.
This time, put in some effort, after you pass, follow through by imitating what a punter does.
After some real effort of trying to get inside his house, he was successful in gaining entry.
Meryl Streep has turned herself, with some effort and after a fair few duds, into a performer who can raise a laugh, while Robert De Niro has by now spent more time spoofing his persona than exploring it.
