To create another stage that amplifies the output signal with a gain $A_v = 3$ using an additional NPN transistor, you can consider adding a common emitter amplifier stage, as it is capable of providing considerable voltage gain. I will assume that the output of your current circuit is taken from the collector of Q1 (Designator: Q1), and you want to feed this into the new stage.

For the sake of simplicity and to maintain consistency with your project's components, I will suggest components similar to those you've already used. I'll introduce a new NPN transistor (Q2), similar to Q1, with the same beta value for consistency. The gain $A_v$ of a common emitter amplifier is approximately given by $-R_C / R_E$, where $R_C$ is the collector resistor and $R_E$ is the emitter resistor. To achieve a gain of 3, you can choose values accordingly, considering $R_C$ to be thrice the value of $R_E$.

This configuration should amplify the signal from Q1's collector by approximately a factor of 3. Adjust the resistor values as needed for your specific application, considering the biasing requirements and the desired operating point for Q2.

To achieve an amplifier circuit with a gain ($A_v$) of 3 using an NPN transistor ($Q2$, assuming you’re adding a new transistor to your circuit, with a $\beta$ (or $h_{FE}$) value of 220 as specified), you will need to design a simple common-emitter amplifier configuration. The gain of the common emitter amplifier can be approximately given by the formula:

where $A_v$ is the voltage gain (note: the negative sign indicates a 180-degree phase shift between input and output), $R_{C}$ is the collector resistor, and $R_{E}$ is the emitter resistor.

For a gain of 3 ($A_v = 3$), you can choose values for $R_{C}$ and $R_{E}$ that satisfy the relation $R_{C} = 3 \times R_{E}$. Let's assume a reasonable value for $R_{E}$ to be $1k\Omega$, then $R_{C}$ should be $3k\Omega$ to achieve the desired gain.

Assuming Vcc (power supply) is adequate and not specified, typically $5V$ or $9V$ can be used for small signal amplifiers.

To ensure proper operation of the transistor in its active region, you may need to add a biasing network at the base of Q2, typically consisting of two resistors forming a voltage divider. However, since the focus here is on achieving a specific gain, and assuming your signal source from C2 is already at an appropriate DC bias level for Q2's operation, this detail can be adjusted as necessary based on your actual supply voltage and desired operating point.

Remember, these are theoretical values and ideal situations. Real-world components and behavior may necessitate adjustments to these values for optimal performance. This design also assumes that the transistor's $\beta$ does not significantly affect circuit performance, which is a valid assumption for many applications but should be verified in high-precision contexts.