Before getting into the tips and tricks, I would suggest stepping back to get clear on your narrative. This may sound a bit fluffy but the "how" is often a function of the "who" and "why".
What I mean by that is you've got to make clear who you want to work for, why you want to work for them and what you have to offer. Given that Google receives over 3 million applications for around 7,000 jobs, taking an unfocused, scattergun approach won't get you far.
Looking at what you've said, you've obviously got some very useful skills, but those skills could equally be applicable to working for an auto dealer, a pharmacy or a funeral parlor (Ok, I'm stretching it a bit but hopefully you get where I'm coming from). In other words, you've got to be able to frame the skills in the context of the firm and role you're looking for.
There are many talented people out there with the right skills, but you've got to be able to answer "Why us?" (What is it about the firm's culture, standing, products do you like?), "Why you?" (You've clearly got the skills but what makes you different to the ten skilled others that have emailed me today? How will you fit into our culture?) and "Why this role?" (Why don't you want to use these skills in a funeral parlor...?)
I always tell clients that once you've properly got your story right, then you can start looking at strategies. Because you are clear on the company culture, you'll get a better sense of what communication channel is appropriate. Because you are clear on the role, you'll get a better sense of what type of individual to speak to. Look at existing roles on job boards to get a better understanding of what they are looking for. Also check out Glassdoor for how people talk about the company.
You have to be strategic about this and I always focus on creating a plan. Keep in mind that companies would far rather not pay for recruiting and you are far more likely to get a foot in the door through a personal introduction. So you can think about alumni associations / LinkedIn groups / Facebook groups / friends of friends / Chambers of Commerce etc etc etc. There are plenty of ways to connect and the warmer the lead the better - connecting blindly via LinkedIn / email doesn't go well... And once you've connected then you can do an informational interview etc - i.e. purely use the meetup to get information on the organisation and not to ask for a job. This would include getting a feel for where your skills would fit into the company.
Finally (or maybe this should be firstly), you must, must, must get your resume / LinkedIn profile sorted. You have to be primed for that first time you reach out to people. Your resume has to be optimized in a way that reflects your target audience, not where you are now.
So that means aligning keywords, job titles etc etc. But all of that is another story...
In essence, you've got to be strategic rather than piecemeal about it. It may not be a quick outcome but at least it will be focused.
Hope this helps.