It came as one SNP MP, Amy Callaghan, issued a “wholehearted” apology after she was heard voicing her support for Mr Grady in a leaked recording.
The staff member who filed a complaint against Mr Grady said it was now difficult to “envision any route back to work”.
Mr Grady, the MP for Glasgow North, was found by an independent investigation to have behaved inappropriately towards the staffer, who was then 19, at a party function in 2016 and was suspended from Parliament for two days.
The SNP withdrew the whip from the MP until he had served his suspension.
But its handling of the complaint has attracted criticism, most notably from the victim himself, who said he had been made to feel as though it was his fault and his life had been made a “living hell".
Over the weekend, audio was leaked to the Daily Mail of a meeting of the SNP group at Westminster, in which Mr Blackford could be heard saying he was “very much looking forward to welcoming Patrick back into the group next week”, and encouraging fellow MPs to offer “as much support as possible”.
Ms Callaghan also urged MPs to rally around Mr Grady.
Following the leak, the SNP threatened its own MPs with criminal action.
Ms Callaghan, the party's pensions spokeswoman, said she was “truly sorry” for the remarks.
She called for a “full root-and-branch review” of the SNP’s internal misconduct and harassment structures.
In a statement, she said: “There is a complainant who has been let down this week by my words and by my party.
"There is no moral high ground or false equivalence when it comes to misconduct. We must do better, myself included.”
The complainer, who still works for the party, told LBC: “In terms of work, the situation is very difficult right now.
“It’s difficult to find any route back to work, to envision any route back to work.
“Especially given the man that’s responsible for staff, Ian Blackford, has directed the MPs to support Patrick.
“How is that going to impact on me should I return back to work? I don’t think I’m going to be well received at work, and I don’t think anyone would blame me for having that view after hearing the audio recording.”
He added: “In terms of work, I’m considering my options, if that includes legal action, then so be it, but I am considering my options at the moment.”
Joanna Cherry, the high-profile SNP MP, said the party had “significant problems” in how it handles complaints.
She tweeted: “I wasn’t at the SNP Westminster group meeting last week.
“I don’t condone the covert recording or leak. However, for some time the SNP has had significant problems in how it handles complaints.
“My party needs to reflect on the contrast between the treatment of different offenders and to review our arrangements for the pastoral care of complainers.”
Ms Cherry has frequently been at odds with party leadership over a number of issues, including her opposition to reform of gender recognition legislation.
Mr Lamont wrote to Mr Blackford over a year ago asking what actions were being taken to prevent sexual misconduct, but did not receive a reply.
The Berwickshire, Roxburgh and Selkirk MP said: “Ian Blackford must break his silence and explain why he has taken no action to prevent sexual harassment in the SNP Westminster group.
“Instead of acting as we requested more than a year ago, he has treated a disgraced MP with more sympathy than the victim who suffered harassment.
“But Ian Blackford has not only failed this victim, his disgusting actions have made it less likely that any future victims come forward.
“This horrible situation has been made even more ugly by the fact the SNP Westminster group is more focused on preventing leaks than preventing sexual harassment.
"Party discipline is more important to them than supporting a victim.
“The cover-up-and-close-ranks culture at the heart of the SNP Westminster group must end. That will only happen if Ian Blackford steps aside.
"He has betrayed a victim and that is unforgivable."
SNP constitution secretary Angus Robertson, a former Westminster leader, accused opposition parties of “throwing bricks while sitting in glass houses”.
He said the outcomes of independent investigations should be respected.
“We need context in this – absolutely, first thoughts are with anybody who has been subject to inappropriate behaviour,” he told STV.
“Secondly, if you are going to have independent processes to look at them and to work out what penalties should be paid, then one should respect those and that’s exactly what’s happening.
“And Ian Blackford is right to recognise that, but at the same time say when somebody has paid the price that they’ve been told that they have to pay, people have to have a future beyond that.”
Mr Grady previously said he was “profoundly sorry” for his behaviour and deeply regretted his actions.
The Glasgow North MP “made an unwanted sexual advance to the complainant that included the touching and stroking of the complainant’s neck, hair and back”, a report into his conduct said.
Labour shadow secretary Ian Murray accused the SNP of a cover-up.
He said: “The leadership of the SNP are spending more time trying to find one of their own who has exposed their duplicity rather than dealing with the issue.
“Political leaders need to set an example against sexual harassment, but the SNP have decided that example should be to cover things up and go after those that have exposed them.”
Lib Dem chief whip Wendy Chamberlain added: “The words of the complainer in this case speak volumes.
“The priority for all parties must be to protect and support whistleblowers and survivors.
“The SNP are clearly not focused on doing that here.”
Speaking on Thursday, First Minister Nicola Sturgeon said: “One of the things that is so important here is that when people suffer this kind of behaviour, that the process they go through should not make the experience or the trauma that they have suffered worse.
“If any complainer feels that is the case, then whatever organisation that has been at stake here, they have to reflect on that and I give an absolute assurance that the SNP will.”