Here's the strategies I'm not sure that you've tried...
1 - Get Promoted Within
It is, by far, easier to be promoted in a company where you are already known and trusted. This usually starts with the work you've already done - supervising the work of lower level employees on a temporary basis, or owning the completion of key areas of work - even if you are not in charge of anyone. Usually you need to demonstrate both timely and high quality completion of work as well as a skill at working with people and having collaborative, positive relationships. Also - showing good judgement in when and where to ask for help.
If you haven't gotten feedback on your work in this area, and in particular what your boss thinks you can do better - get it. Understanding your weak points (everyone has some!) is key to working toward this goal.
Also - have a frank discussion with your management on do they see this role being available to you in the near term. If your company isn't growing, then the internal route may not be possible. If you don't get a straight answer, take a look at the company. If it's not growing in headcount and all the managers have a lot of experience and are no where near retirement, there may not be any opportunity here.
2 - Look for lead positions rather than manager positions.
In many cases, a first time management position may be described as "lead" rather than "manager" - a lead is often someone who has authority over the completion of the work, but is only an advisor on the performance of the staff (as in hiring, firing, career development and performance reviews of direct reports). It's a bit easier to grow into management when you start as a lead.
There is a difficult catch 22 in the range of having the supervisory responsibility for the performance of other employees. At least in the US, this where the lawsuits start. If the manager isn't ethical, responsible and respectful then he can open the company up to lawsuits. If the manager isn't able to comprehend and work within the company culture - then his direct reports may not thrive. If he's simply a jerk, the company risks loosing good staff. It's a lot of trust to put in a stranger.
A lead position is usually easier to acquire because the lawsuit/trust/risk realm falls on the manager. You may well be able to negotiate a lead position where you and your new employer recognize the goal of you becoming a manager, and if they love you enough, they may be willing to make an offer with a promotion plan in place to get you to a manager position provided you can meet their defined expectations.
3 - Network, network, network
Because management positions are so dependent on soft skills and trust, they are often also as much a matter of networking as interviewing prowess. There's a point in leadership careers where you are being hired for who you are more than what you can do. If you've had a depth of leadership experience that is not supervisory experience, then you should be in a position where you have a pretty wide personal network of people who have had lots of positive experience being led by you. Assuming they aren't all working with you now - reach out to colleagues in other companies who would recommend you as a leader and see what opportunities they would recommend and support you for.
As a litmus test, if you are reading this and thinking "I can't do this, my professional network is entirely within my current company" - then I would say you don't have enough leadership experience for a management role. Build that experience by taking on leadership roles that give you opportunities to collaborate with people unlike yourself - including in other parts of the company, and in other companies (either as vendors, partners or customers of your company).
Manage People in a Low Risk Environment
Volunteer groups of all sorts are always looking for good leaders. And in all honesty, I find volunteer leadership to be harder than paid professional leadership, because the motivations of your reports will be far more diverse, and their skill sets can be all over the map. If you can manage a team of volunteers doing something important, you have some GREAT management experience you can put on your resume.
Another Sad Truth
I've applied to numerous first-level management positions myself, and also interviewed candidates for such levels. I have to say - wow is there a lot of competition! In the roles I've interviewed, there have usually been a daunting number of good candidates for a single position - and most if not all of them have previous management experience. I've also spoken to several people in the tech industry who have taken all sort of level demotions to continue their management careers across companies (for example, a director becoming a manager, or a VP becoming a director). Some of this is that different companies will rate the same basic job functions differently - a VP of a small company may manage 25 direct reports with a small budget, while a Senior Manager of a big company does the same thing. But in other cases it's because the applicant just wanted to move to the company and they were willing to take a demotion of both role and responsibility to do so.
So.. why should a company hire someone with no management experience when they can hire a great candidate with excessive experience? Companies will select the best person they think they are likely to get... and in this case, many know that they can get someone with huge experience quite quickly.