I very much doubt reading POH would be enough as a 'familiarization' training. You will need endorsement for a type (in other words flight instruction in the type) and often other additional practical obstacles would stand in the way (like flying club or insurance requirements). If it was so easy to move within SEP(L) someone could fly 172 and then after 'familiarization' jump into say Piper Mirage or Cessna 210.
Oh my gosh! I understand that there are some regulatory requirements which may drive this under some authorities, but let's not lose sight of the fact that nearly all of the GA fleet of certified piston single engine aircraft (excepting a few pressurized exotic types) are designed and certified for a competent pilot to READ THE FLIGHT MANUAL and then fly. That's not to say that familiarization training is a bad thing, but the safety of flight for competent pilot does not depend upon it. It is a certification requirement that all of these aircraft NOT require unusual pilot skill or attention - they are not hard to fly, and do not bite a pilot exercising appropriate skill and attention. This premise does not stop at light certified aircraft, even very big aircraft can be flown safely by a pilot who has familiarized them self with the operation - but I agree that opportunity is uncommon.
When you fly a certified plane, you're not the test pilot, that's been done for you already. The test pilot has demonstrated that you don't need to have test pilot skills to fly it! A part of the appropriate "skill and attention" would be to recognize that you're "new" in a type, and take things slowly, without immersing yourself in challenging conditions for the first while. And, keep the flight manual close at hand, and follow checklists found in that manual.
Many times, I have jumped into something I have never flown before, read the flight manual, and off I went. Other times, while flying a modified aircraft, I've flown something which had never flown before, and written a flight manual supplement for it. The only type which tried to bite me was actually not Canadian type certified, and now I know why. Other than that, apply yourself, and follow the instructions.
Insurance companies and regulators can tend to make a fuss out of this, particularly for lower time pilots. Flying clubs can exacerbate this - unfairly to those pilots. By all means, obtain the required training, but recognize that you might be doing that to meet more of a regulator requirement than an actual concern about demonstrated skills.
To cite the example. if you can fly a 172 well, and read and follow instructions well, you could safely check yourself out in a C210, if you give yourself some room, and take it easy for the first while.... Have confidence in yourself - and the test pilot who certified it as not requiring unusual pilot skill and attention!