Other Ways to Progress Without Increasing Weight
It is a well-known law for muscle building: to keep muscle growth consistent over time, you MUST progressively overload. This means that you should be always challenging your body and making it harder for yourself. If you keep your workouts at the same intensity for too long (let's say almost a month), then your body will have adapted to this intensity and any further growth will not be seen as necessary by your muscles. To prevent this plateau, it is vital that you regularly take it up a notch. This is mostly done by stacking more weight onto your exercises, which is a very valid and efficient method. However, if you want to explore other options, perhaps because of injury, then you should give something else a go. As I am still recovering from my back injury, I have used all of these to substitute for loading my spine too much.
Increase the sets/reps
This one is pretty straight-forward: more work equates to more muscle tearing. But, that doesn't mean you should fatigue yourself on every set of every exercise, as this will overdo it and leave you exhausted before you've even gotten started.
Add a resistance band
A resistance band can be used to accompany an exercise (like placing it around your knees as you squat) or replace the weight altogether (perhaps in place of a dumbbell on a bicep curl). Either way, it works to keep constant tension all throughout the rep, and/or activate more neglected muscles in the movement.
Increase the duration of a rep
Similar to increasing the amount of reps/sets, you are putting your body under stress for longer, tearing more fibres. You may also be hitting different length fibres, which will definitely leave you aching the next morning! This is also a great way to practice your form, as you can't use your momentum to swing the weight.
Shorten your rest period
Resting between sets allows for short term recovery so that you can still complete the next set, with sufficient form of course. Limiting your resting time inhibits some of this recovery, ensuring the next set will be even harder to complete.
Focus on the eccentric portion
A rep is split into two portions: eccentric and concentric. When talking about a lat-pull down, for instance, the concentric part would be pulling the weight down, and the eccentric would be guiding it back up to the start position. The eccentric is always the easier portion, as it doesn't require your muscles to suddenly contract. Therefore, by slowing down and controlling how you return to the start position, you are adding more challenge to the exercise.
Pauses can be BRUTAL on some exercises, and they reinforce a high level of control so I would confidently recommend adding them to your routine. Imagine you are performing a hip thrust: you can pause halfway and hold for a second, or pause twice, or three times. You can focus on your weakest distance using this method, whether that's from the bottom to a third of the way, or just towards the end. Pausing at the top and squeezing your muscles is highly effective too. I would advise using this method on exercises with a larger range of motion to maximise its efficacy.
Practice depth/fully stretch
To get the most out of an exercise, it can work best when starting at a fully stretched position. This means that your muscle is lengthened as much as it can go. On a laying hamstring curl, this point would be when your legs are straight. Though this makes use of the full range of motion, it may result in losing tension in the movement; your target muscles will have a short moment of rest. As well as this, mobility restrictions could get in the way. When at the bottom of a squat, for example, the quads and glutes are most stretched, but if you have poor hip or ankle mobility, you may struggle to reach this position.
The reason all of these are effective is because they induce metabolic stress; they all make an exercise harder for your muscles. Muscle growth comes from challenging yourself constantly, supported by a substantial consumption of calories and protein. As long as you abide by these guidelines, you will see some incredible stuff happen!