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Sites like linkedin.com are great as it's a social network site just for professionals to maintain business leads. There are always a lot of IT positions up there.
You don't necessarily need a large company. There are lots of medium sized companies which are in dire need to QA staff. Especially here in the states as we have a lot of regulations now which require lots of paperwork process to be in place.
The best way to find out what people in your area are looking for is to post your info up on monster.com and dice.com and get a few interviews. Even if you don't like the company taking the interview will give you a great deal of insight into what the tools and processes are which they are looking for. You can then bone up on those tools and processes and then land the position at the company which is right for you.
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10 years in software QA itself makes you a specialized professional. As by your detail, I assume that it has not only been QA for all 10 years, initially probably you were in requirements study, then to project management and finally to testing/QA. The experience tells that you have conducted mostly manual testing and especially functional testing. No tools or a standard methodology (CMMI, Scrum/Agile etc.) have been encountered during your journey. First of all forget that you have to start afresh, but you have to carry along with the same journey that you have done so far. it has to be further enhancement/upgradation in your skill sets, and probably an exposure to the hidden world (so far to you!) of QA. Post your resumes – there are many platforms for that – but prior to you set your goals in mind, as it seems it is still not clear what exactly you want. Once it is clear, talk to your current organization, that what enhancements or improvements you want in QA and show them on paper what could be the benefits out of it. Assure them about the benefits you perceive to draw out of it.
It is not necessary that QA in large organizations will be always in a better shape than the small/medium organizations. It all depends on internal policies and the people who drive the organization to new heights. So first, give a try at the place where you are already, get out of this burnout mode and recharge yourself for your enhancement.
Software Quality, Software Quality Assurance, Jobs for IT professionals
I've been in Software QA for about 10 years. I've only worked in startups (5-15 people), or small groups in large corporations where I've been the only QA person. I never had a QA mentor, so I learned as much as I could by reading books and going online and applying whatever I could (in the time allotted). Also, I've never had formal QA training as the companies couldn't afford to send me (time and money). Over those 10 years I've touch on requirements, project management, testing and test planning. Unfortunately, each of those aspects in the small companies I've been in weren't followed in a structured way. Random processes lead to random results which leads to burnout. I feel it is time for me to move out of the startups and into an area where I am going to learn about best practices. I'm just not sure of the types of companies I should be pursuing. I have a feeling it will be a large company that already has an established QA group. I just don't know where to start. Any suggestions would be greatly appreciated. Thank you. 
