I believe I have found a partial answer to my question. Consider the following: If the first-order part of $RCA^{*}_0$ is $I$$\Delta_{0}$ + $Exp$ + $B$$\Sigma_{1}$, where $B$$\Sigma_{1}$ is the Boundedness Principle for $\Sigma_{1}$ formulas (as Prof. Enayat suggests in his comment to his answer to the mathoverflow question question, "van der Waerden's theorem in Reverse Mathematics" (question 316480), one can use the following theorem of Dimitracopoulos and Paris (mentioned in the paper "Where Pigeonhole Principles Meet Konig Lemmas")

Over $I$$\Delta_{0}$ + $Exp$, $\forall$ $x$($\Sigma_{1}$: $x$ +$1$ $\rightarrow$ $($ 2 $)^{1}_{x}$$)$ is equivalent to $B$$\Sigma_{1}$ for all $n$ $\in$ $\mathbb N$, where $\forall$$ $x$ $($\Sigma_{1}$: $x$ + $1$ $\rightarrow$ $($ 2 $)^{1}_{x}$$)$ is the $\Sigma^{0}_{1}$ Pigeonhole Principle

to substitute the $\Sigma^{0}_{1}$ Pigeonhole Principle for $B$$\Sigma_{1}$ in the first-order part of $RCA^{*}_{0}$, giving the type of pigeonhole principle needed to prove Siegel's Lemma in $EFA$, if it in fact can be....

I believe I have found a partial answer to my question. Consider the following: If the first-order part of $RCA^{*}_0$ is $I$$\Delta_{0}$ + $Exp$ + $B$$\Sigma_{1}$, where $B$$\Sigma_{1}$ is the Boundedness principle for $\Sigma_{1}$ formulas (as Prof. Enayat suggests in his comment to his answer to the mathoverflow question question, "van der Waerden's theorem in Reverse Mathematics" (question 316480)), one can use the following theorem of Dimitracopoulos and Paris (mentioned in the paper "Where Pigeonhole Principles Meet Konig Lemmas")

Over $I$$\Delta_{0}$ + $Exp$, $\forall$ $x$($\Sigma_{1}$: $x$ +$1$ $\rightarrow$ $($ 2 $)^{1}_{x}$$)$ is equivalent to $B$$\Sigma_{1}$, where $\forall$$ $x$ $($\Sigma_{1}$: $x$ + $1$ $\rightarrow$ $($ 2 $)^{1}_{x}$$)$ is the $\Sigma^{0}_{1}$ Pigeonhole Principle

to substitute the $\Sigma^{0}_{1}$ Pigeonhole Principle for $B$$\Sigma_{1}$ in the first-order part of $RCA^{*}_{0}$, giving the type of pigeonhole principle needed to prove Siegel's Lemma in $EFA$, if it in fact can be....

I believe I have found a partial answer to my question. Consider the following: If the first-order part of $RCA^{*}_0$ is $I$$\Delta_{0}$ + $Exp$ + $B$$\Sigma_{1}$, where $B$$\Sigma_{1}$ is the Boundedness Principle for $\Sigma_{1}$ formulas (as Prof. Enayat suggests in his comment to his answer to the mathoverflow question question, "van der Waerden's theorem in Reverse Mathematics" (question 316480), one can use the following theorem of Dimitracopoulos and Paris (mentioned in the paper "Where Pigeonhole Principles Meet Konig Lemmas")

Over $I$$\Delta_{0}$ + $Exp$, $\forall$ $x$($\Sigma_{x+1}$: $x$ +$1$ $\rightarrow$ $($ 2 $)^{1}_{x}$$)$ is equivalent to $B$$\Sigma_{n + 1}$ for all $n$ $\in$ $\mathbb N$, where $\forall$$ $x$ $($\Sigma_{n + 1}$: $x$ + $1$ $\rightarrow$ $($ 2 $)^{1}_{x}$$)$ is the $\Sigma^{0}_{1}$ Pigeonhole Principle

to substitute the $\Sigma^{0}_{1}$ Pigeonhole Principle for $B$$\Sigma_{1}$ in the first-order part of $RCA^{*}_{0}$, giving the type of pigeonhole principle needed to prove Siegel's Lemma in $EFA$, if it in fact can be....

I believe I have found a partial answer to my question. Consider the following: If the first-order part of $RCA^{*}_0$ is $I$$\Delta_{0}$ + $Exp$ + $B$$\Sigma_{1}$, where $B$$\Sigma_{1}$ is the Boundedness Principle for $\Sigma_{1}$ formulas (as Prof. Enayat suggests in his comment to his answer to the mathoverflow question question, "van der Waerden's theorem in Reverse Mathematics" (question 316480), one can use the following theorem of Dimitracopoulos and Paris (mentioned in the paper "Where Pigeonhole Principles Meet Konig Lemmas")

Over $I$$\Delta_{0}$ + $Exp$, $\forall$ $x$($\Sigma_{1}$: $x$ +$1$ $\rightarrow$ $($ 2 $)^{1}_{x}$$)$ is equivalent to $B$$\Sigma_{1}$ for all $n$ $\in$ $\mathbb N$, where $\forall$$ $x$ $($\Sigma_{1}$: $x$ + $1$ $\rightarrow$ $($ 2 $)^{1}_{x}$$)$ is the $\Sigma^{0}_{1}$ Pigeonhole Principle

to substitute the $\Sigma^{0}_{1}$ Pigeonhole Principle for $B$$\Sigma_{1}$ in the first-order part of $RCA^{*}_{0}$, giving the type of pigeonhole principle needed to prove Siegel's Lemma in $EFA$, if it in fact can be....