In practice, the apparatus measuring the spin should be localized somewhere in space (it cannot fill the whole universe!) and this fact implies that you always make a measurement of position (actually very rough in general), even if you are measuring the spin. Suppose that $\Omega \subset R^3$ is the bounded region in $R^3$ where the apparatus is localized.
The simplest (naive) mathematical model of the apparatus I could imagine is the following.
The YES-NO observable associated with the apparatus measuring, say, if the spin is directed along z+, has the form of the orthogonal projector: $$P_{\Omega} \otimes P_{z+}$$
Here
$P_{z^+} = |z+\rangle \langle z+|$ is the obvious projector in $C^2$ along the states with spin $z+$-directed ,
whereas $P_\Omega$ is the operator (orthogonal projector in $L^2(R^3)$)
$$(P_\Omega \psi)(x) = \chi_\Omega (x) \psi(x)\:.$$
This observable admits two values (its eigenvalues) $0=$ NO and $1=$YES. YES means that the particle is found in $\Omega$ AND the spin is found to be directed along $z+$.
NO means that the the particle is not found in $\Omega$ OR the spin is not along $z+$.
There is another elementary YES-NO observable associated with the spin detected along the direction $-z$, with analogous meaning. It is the orthogonal projector:
$$P_{\Omega} \otimes P_{z-}\:.$$
The observable associated with the spin along $z$ --referring to this experiment-- is not the standard operator $S_z = \sigma_z/2$ (I am assuming $\hbar =1$).
It is instead constructed, via spectral decomposition, taking the above elementary observables (projectors) into account and combining them with the corresponding values of the spin (which turn out to be the eigenvalues of the overall observable).
$$\Gamma_{z,\Omega} = \frac{1}{2}P_{\Omega} \otimes P_{z+} - \frac{1}{2}
P_{\Omega} \otimes P_{z-} = P_\Omega \otimes S_z\:.$$
You see that it includes a rough measurement of the position: it just checks if the position of the particle is in $\Omega$.
To measure the spin of the particle the supports of the components $\phi_i$ must have a non-negligible intersection with $\Omega$. In general the measurement procedure of the spin, for instance along $z+$, even affects the surviving component $\phi_{+1/2}$.
You see that, only if the support of $\phi_{+1/2}$ is completely included in $\Omega$, the wavefunction is not affected by the measurement of the spin, otherwise, after the procedure (supposing to have found spin $+1/2$), the state, up to a normalization constant, is described by:
$$P_\Omega \phi_{+1/2} \otimes |z+\rangle \:.$$
It is questionable if we have defined an observable $\Gamma_{z,\Omega}$ in that way. The point is that $\Gamma_{z,\Omega}$ admits a third eigenvalue, $0$, associated with
the projector $P_{R^3-\Omega} \otimes I$. Actually, there is no real measurement in the region $R^3-\Omega$, since we are not assuming that there are detectors therein. We are simply using the argument: "if the particle is not found in $\Omega$ it must be found outside it". For several reasons I am always a bit suspicious to this sort of formal arguments. A more physically safe interpretation could be that we are performing a conditioned measurement of the spin. $P_\Omega$ is nothing but a filter: only the particles which pass through it are measured.
