How to calculate the variation of enthalpy in a reaction that does not occur in standard conditions?
For a pure substance you can write generally that the enthalpy depends on pressure as follows:

$$\left(\frac{\partial H}{\partial p}\right)_T = V - T \left(\frac{\partial V}{\partial T}\right)_p \tag{1}$$

If you evaluate this for an ideal gas you find that  $\left(\frac{\partial H}{\partial p}\right)_T = 0$. If the substance is present in a condensed phase with a density (or molar volume) that is weakly dependent on the temperature, then roughly  

$$\left(\frac{\partial H}{\partial p}\right)_T = V$$

If you speak of standard enthalpies then by definition there is no pressure dependence, as they are defined at a particular pressure. 

If your reaction involves gases which can be assumed ideal, then you can use the standard enthalpy for all reagents and products, since according to my answer the enthalpy of an ideal gas is independent of pressure. On the other hand if it involves condensed phases you need to evaluate the molar volume and perhaps also the thermal expansion coefficient $\alpha$, since Eq. (1) can be rewritten as

$$\left(\frac{\partial H}{\partial p}\right)_T = V (1- T\alpha)$$

For the enthalpy of reaction you might write

$$\left(\frac{\partial \Delta_r H}{\partial p}\right)_T = \sum_i \nu_i V_{m,i} (1- T\alpha_i)$$

where $\nu_i$ are stoichiometric coefficients. In the summation in the rhs account should be taken for sign changes depending on whether species are reactants or products.

If you want to evaluate the enthalpy of reaction at one pressure given the value is known at another pressure, and the molar volumes are known, you can integrate the previous equation, giving (at constant T)

$$\Delta_r H(p)  =  \Delta_r H(p^\circ) + \int_{p^\circ}^{p} \sum \nu_i V_{m ,i}(1- T\alpha_i) dp$$

where $p^\circ$ is the reference pressure at which the enthalpy is already known. This last expression may be simplified in various ways depending on the state of the species and the pressure integration range. If the molar volumes and expansion coefficients are approximately constant in the pressure range then  

$$\Delta_r H(p)  =  \Delta_r H(p^\circ) + (p -p^\circ) \sum \nu_i V_{m,i} (1- T\alpha_i)$$

If the species have negligible thermal expansion coefficients then

$$\Delta_r H(p)  =  \Delta_r H(p^\circ) + (p -p^\circ)\sum \nu_i V_{m,i}$$

and so on. For instance, for a reaction converting a solid between two forms $\text{react}$ and $\text{prod}$ having constant molar volume and negligible thermal expansion coefficients in the pressure range $p^\circ$ to $p$, 

$$\ce{\text{react}&lt;=&gt;\text{prod}}$$

$$\Delta_r H(p)  =  \Delta_r H(p^\circ) + (p -p^\circ)(V_m(\text{prod})-V_m(\text{react}))$$