Can you do hypertrophy training after strength training?Short Answer: Of course.
The two are interdependent, not mutually exclusive.
Meaning you will rely on developing a certain amount of strength over time to develop hypertrophy and you will need a certain amount of volume (weight x sets x reps - typically associated with hypertrophy) and time under tension to improve strength.
You could focus on either in any order and they are in fact typically quite complementary.
For instance you could put them in the same training session even:
A1) 3x5 Back Squat
A2) 3x5 Loaded Chin Ups
B1) 3x10 Front Squat
B2) 3x12 Assisted Pull Up
I know...mind blown...
To some extent strength training is the same as resistance training is the same as weight training.
The only thing that really changes between training for strength and training for hypertrophy is where they fall on a relative spectrum of overall volume.
There are also different types of hypertrophy. See my answers to Is it better to lift light weights than heavy ones? What about for weight loss, or muscle 'toning,' which is better? and Is it better to keep lifting weights until failure or not?
The potential for hypertrophy with an exclusive focus on strength (<6 reps) is slower for progress and potentially smaller because you're never stimulating the smaller but still hypertrophy capable type 1 muscle fibers.
In essence you're never tapping the full hypertrophic spectrum of muscle fiber stimulation and sarcoplasmic hypertrophy capability lifting in an exclusively strength focused situation.
You'll still get hypertrophy and it will still be considerably more than if you trained an exclusive focus on muscular endurance (>12 reps) because the potential of type 2 fibers is still significantly greater than type 1 fibers.
That's part of the reason why many people will still get hypertrophy on a program such as Stronglifts 5x5 or Starting Strength. Other reasons being that beginners adapt far more quickly than intermediate or advanced lifters, and volume is often high enough for hypertrophy to occur.
The short answer being: It's complicated.
It's a spectrum.
You can certainly get a lot of hypertrophy with a great deal of focus on strength and vice versa (lots of strength even with a focus on hypertrophy) it's just a lower upper limit respectively in each case. You'll only gain so much strength using traditional hypertrophy rep ranges (6-12) and you'll only gain some much hypertrophy using traditional strength rep ranges (<6).
I'd say the optimal path is to use an integrated system no matter what your end objectives may be. Powerlifters will lift for strength in their core lifts but then supplement a great deal of their training with traditionally hypertrophy oriented rep ranges.
Likewise bodybuilders (good ones anyway...) will periodically lift for strength to increase the amount of weight they can use for certain phases of training so that they can lift more in the traditionally hypertrophy rep ranges.
I'm very fond of what's called 'Alternating Periodization.' Whereby you have a phase of accumulation (volume - more traditionally hypertrophy oriented training like 4x8 or 4x6 or 3x10 etc...) for 3-6 weeks, and a phase of intensification (strength/explosive power - more traditionally strength oriented training like 6x3, 5x5, 5/3/1, etc...).
It's a simplistic integrated approach but not as simplistic as something like SL5x5 or SS (I feel the appeal of these programs is rooted in their simplicity but that doesn't necessarily make them 'more effective').
Ultimately you should decide what your end objectives are and formulate a plan based around those end objectives.
If it's strength then working with a program like SL5x5 or SS is probably more appropriate for a beginner anyway. However as you progress you'll probably find your way into more advanced strength training systems like 5/3/1, Juggernaut, Cube or Westside. Why? There's an upper limit of how much strength you can gain on simplistic programs...
If it's hypertrophy (and judging by the question details this seems to be the case) then you have a little more forgiveness in terms of programming. Scrawy to Brawny, Westside for Skinny Bastards, AMD, Built for Show, EDT, etc... There are WAY more programs out there that focus on hypertrophy it seems. Hypertrophy appears to thrive on my variance and research indicates that although a 6-12 rep range is probably ideal for most of your training volume to fall for hypertrophy, a broad spectrum of stimulation seems to be most effective so some strength training and some endurance training thrown into the mix to enhance total muscularity.
However, I'd say most people are better off getting as strong as they can first before switching to a hypertrophy focus (yes - you could use a program like SL5x5 or SS first). Why? In my experience getting strong first is more important because it increases your tolerable loads in that 6-12 range later. You need maximum stimulation and being strong facilitates that better as many the world class bodybuilder could attest to. Arnold was a powerlifter before he was a bodybuilder after all.
Too often I've seen frustrated beginners/intermediates who are following body part splits but aren't seeing results because they quite literally just can't hit the right intensity. Nor can they typically hit enough frequency of stimulation. Doing strength training first with a focus on compound movements done frequently (2-4x a week for the same body parts) seems most effective in getting them to a level where body part splits common in bodybuilding can now be utilized effectively.
Without a baseline of strength first, utilizing a bodybuilding program just isn't as effective.
Whereas getting to an intermediate level on these lifts (Killustrated Weightlifting Standards) first will probably do a better job at leading to hypertrophy later in your training.
Vice versa, it's been my experience that starting with more of a bodybuilding approach does not adequately translate to strength improvements in the long run either. The waterfalls in the one direction, but not the other, though you can and will improve strength to a certain degree even with higher rep ranges it won't be nearly as much.
Better to go strength first and hypertrophy second or even congruently. You could take the above example for instance and I believe potentially make a program like SL5x5 or SS even better by adding additional volume after the main or core lifts are done. Or at least integrate a cycle of greater overall volume every four weeks within the program.
i.e. After you finish Squatting, Benching, Deadlifting in SS, add 3-6 assistance or accessory exercises that focus more on a 6-12 rep range to improve areas you wish to improve. Do the regular SS program for four weeks, then a modified version of SS with the additional volume for 4 weeks and alternate them back and forth. Again this is called Alternating Periodization.
I don't think SL5x5 vs SS is really that big a difference even accounting for less volume found in SS. Choose one stick to it for a while, then maybe even consider the other. Give each one 3-4 months to work first before you switch though.