Let $\delta$ be 3 month and consider points of interest $\{T_i\}_i$ evenly spaced with $T_{i+1} -T_i = 3 month$. The Forward Rate $F_m^n(t)$ from period m to n at time $t$ is defined by $$(1 + \delta (n-m) F_m^n(t)) = \frac{B(t,T_m)}{B(t,T_n)},$$ where $B(t,T_i)$ is the time $t$ value of a zero coupon bond that matures in $T_i$.
A swap rate $S_m^n(t)$ a time $t$ of a swap starting in $T_m$ and ending in $T_n$ can be written as
$$ S_m^n(t)=\sum_{i=m}^{n-1} \frac{\delta B(t,T_{i+1})}{\sum_{j=m}^{n-1}\delta B(t,T_{j+1})}F_i^{i+1}(t) $$
It holds
$$F_0^1(0) = (S_0^2(0) - \frac{1}{2+\delta F_1^2(0)}F_1^2(0))(1+1\frac{1}{1+\delta F_1^2(0)}) $$
Derivation
Use definition for Swap Rate
$$ \frac{B(0,T_1)}{B(0,T_1)+B(0,T_2)}F_0^1(0) + \frac{B(0,T_2)}{B(0,T_1)+B(0,T_2)}F_1^2(0) = S_0^2(0)$$
Now use
$$ (1 + \delta F_0^1(0))(1+\delta F_1^2(0)) = \frac{1}{B(0,T_2)} $$
and
$$ (1 + \delta F_0^1(0)) = \frac{1}{B(0,T_1)}$$
which leads to
$$ \frac{\frac{1}{(1 + \delta F_0^1(0))}}{\frac{1}{(1 + \delta F_0^1(0))} + \frac{1}{(1 + \delta F_0^1(0))(1+\delta F_1^2(0))}}F_0^1(0) + \frac{\frac{1}{(1 + \delta F_0^1(0))(1+\delta F_1^2(0))}}{\frac{1}{(1 + \delta F_0^1(0))} + \frac{1}{(1 + \delta F_0^1(0))(1+\delta F_1^2(0))}}F_1^2(0) = S_0^2(0)\\ \Leftrightarrow \frac{1}{1+\delta F_1^2(0)} F_0^1(0) + \frac{1}{2+\delta F_1^2(0)}F_1^2(0) = S_0^2(0) $$
which is equivalent to my the expression above.